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The real Plücker–Klein map V. A. Krasnov P.G. Demidov Yaroslavl State University
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     IntroductionThe intersection of two complex quadrics is called a biquadric. If we mark a non-singular quadric in the pencil of quadrics through a given biquadric, then the given biquadric is called a marked biquadric. In the classical papers of Plücker and Klein (see [1], [2]), a Kummer quartic in three-dimensional projective space was canonically associated with every three-dimensional marked biquadric (that is, with a quadratic line complex whose Plücker–Klein quadric is marked). This yields a map from the set of all three-dimensional marked biquadrics to the set of Kummer quartics. We call it the classical Plücker–Klein map. This map was considered in detail in the classical treatises [3], [4]. Rohn [5] studied the real Plücker–Klein map from the set of all real quadratic line complexes to the set of real Kummer quartics. Since the necessary notions of real algebraic geometry did not exist in the second half of the nineteenth century, statements of the results in [5] are inaccurate. A brief exposition of Rohn’s results was given in the treatise [4] (beginning of the twentieth century), but the statements are again inaccurate. This was mentioned by Barth in his preface to a new edition of [4]. The study of the classical Plücker–Klein map was continued in a series of my papers (see [6]–[11]), which in particular contain more accurate statements of some of Rohn’s results. The classical Plücker–Klein map was generalized in Reid’s thesis [12] devoted to biquadrics. To every non-singular intersection of two quadrics (a biquadric) $B\,{=}\,Q_1\,{\cap}\, Q_2\,{\subset}\,\mathbb{P}^{2g+1}$, $g\,{\geqslant}\,2$, there corresponds a Kummer variety $K(B)$ of dimension $g$ in the maximal orthogonal Grassmannian of $Q_1$ (see [12], § 4.1). Reid only constructed the generalized Plücker–Klein map. Reid’s construction was not further developed in other papers devoted to intersections of two quadrics, except for [13], where the complex generalized Plücker–Klein map was studied in detail. In what follows it is referred to just as the Plücker–Klein map. In the present paper we study the Plücker–Klein map for real marked biquadrics of arbitrary odd dimension. Before giving exact statements of the problems posed and the results obtained, we need to give a number of definitions and assertions in [13] concerning the complex Plücker–Klein map. In what follows $V$ is a complex vector space of dimension $n=2g+2\geqslant 6$. Since the group $\operatorname{GL}(V)$ acts transitively on the space of non-degenerate quadratic forms on $V$, we can assume in our study of marked biquadrics that the marked quadric in the pencil of quadrics does not change (that is, regard it as fixed). Thus let $\mathfrak{Q}\subset\mathbb{P}(V)$ be a fixed non-singular quadric and let $\mathfrak{q}(\mathbf{v})$ be a quadratic form on $V$ such that the equation $\mathfrak{q}(\mathbf{v})=0$ determines the quadric $\mathfrak{Q}\subset\mathbb{P}(V)$. Unless otherwise stated, the form $\mathfrak{q}(\mathbf{v})$ is also assumed to be fixed. The scheme intersection $\mathfrak{Q}\cap Q\subset\mathbb{P}(V)$, where $Q\subset\mathbb{P}(V)$ is an arbitrary quadric different from $\mathfrak{Q}$, is called a $\mathfrak{Q}$-biquadric. The set of such biquadrics is denoted by $\operatorname{BQ}=\operatorname{BQ}(\mathfrak{Q})$. We also write $\operatorname{BQ}^\circ=\operatorname{BQ}^\circ(\mathfrak{Q})$ for the subset of non-singular $\mathfrak{Q}$-biquadrics in $\operatorname{BQ}$. In what follows we consider only biquadrics that are $\mathfrak{Q}$-biquadrics. Therefore $\mathfrak{Q}$-biquadrics will be referred to as just biquadrics. Every biquadric $B\in\operatorname{BQ}$ determines a line $L=L(B)$ (a pencil of quadrics) in the projective space $\mathbb{P}(\mathrm{S}^2V^\vee)$ of quadrics. This line consists of all quadrics through $B$. Since the pencil contains the marked quadric $\mathfrak{Q}$, the biquadric $B\in\operatorname{BQ}$ is a marked biquadric. Any biquadric $B\in\operatorname{BQ}$ is uniquely determined by the pencil of quadrics $L=L(B)$ and, therefore, the set $\operatorname{BQ}$ of biquadrics is equal to the set of lines in $\mathbb{P}(\mathrm{S}^2V^\vee)$ through the fixed quadric $\mathfrak{Q}$. Hence $\operatorname{BQ}$ is a projective space of dimension $N$, where and $\operatorname{BQ}^\circ$ is the complement of the discriminant hypersurface in $\operatorname{BQ}$. Therefore $\operatorname{BQ}^\circ$ is an affine algebraic variety of dimension $N$.If $B\in\operatorname{BQ}^\circ$, then the pencil $L(B)$ contains exactly $n$ singular quadrics $Q_1,\dots,Q_n$. Each of them has a unique singular point. Given any $p\in L(B)$, we denote the corresponding quadric by $Q_p$. Let $p_1,\dots,p_n$ be the points of the line $L(B)$ that correspond to the singular quadrics $Q_1,\dots,Q_n$. We also write $\mathfrak{p}\in L(B)$ for the point corresponding to the marked quadric $\mathfrak{Q}$. Let $\Phi=\Phi(\mathfrak{Q})$ be the Fano variety whose points are projective $g$-planes in $\mathfrak{Q}$. It consists of two connected components $\operatorname{G}_\pm=\operatorname{G}_\pm(\mathfrak{Q})$. The components $\operatorname{G}_\pm$ are smooth varieties of dimension $g(g+1)/2$. They are isomorphic. Note that $\operatorname{G}_\pm$ are three-dimensional projective spaces for $n=6$ (see [14], Ch. 6) and six-dimensional quadrics for $n=8$ (see [15], § 20.3). Since $\mathfrak{Q}$ contains no planes of dimension greater than $g$, the varieties $\operatorname{G}_\pm$ are called the maximal orthogonal Grassmannians of $\mathfrak{Q}$. For every $B\in\operatorname{BQ}^\circ$ there are Kummer varieties $K_\pm=K_\pm(B)\subset\operatorname{G}_\pm$ of dimension $g$. Each of them consists of all $g$-planes $\Pi\in\operatorname{G}_\pm$ such that the set-theoretic intersection of $\Pi$ and $B$ is the union of two $(g-1)$-planes (possibly coinciding). In the general case, the varieties $K_\pm(B)$ were introduced in [12]. We call them the Kummer varieties associated with the biquadric $B$. The varieties $K_+(B)$, $K_-(B)$ are isomorphic to the Kummer variety of the Jacobian of the double covering of the line $L(B)$ branched at the points $p_1,\dots,p_n\in L(B)$ (see [12]). We write $\operatorname{KV}_\pm=\operatorname{KV}_\pm(\operatorname{BQ}^\circ)$ for the sets whose elements are the Kummer varieties in $\operatorname{G}_\pm$ arising from biquadrics $B\in\operatorname{BQ}^\circ$. Then there are maps
$$
\begin{equation*}
\operatorname{PK}_\pm\colon \operatorname{BQ}^\circ\to\operatorname{KV}_\pm
\end{equation*}
\notag
$$
which send every biquadric $B$ to the Kummer varieties $K_\pm(B)\subset\operatorname{G}_\pm$. Since $\operatorname{PK}_\pm$ are the classical Plücker–Klein maps for $n=6$, we preserve this terminology for arbitrary $n\geqslant6$. Originally these are maps between sets, but $\operatorname{BQ}^\circ$ is an affine algebraic variety and the fibres of the Plücker–Klein maps are algebraic curves, namely, projective lines punctured at $n$ points and located in $\operatorname{BQ}^\circ$ in a special way. This endows the sets $\operatorname{KV}_\pm$ with the structure of affine algebraic varieties in such a way that the maps
$$
\begin{equation*}
\operatorname{PK}_\pm\colon \operatorname{BQ}^\circ\to\operatorname{KV}_\pm
\end{equation*}
\notag
$$
are regular submersions (see § 1 for details). Let $\operatorname{G}$ be one of the varieties $\operatorname{G}_\pm$, let $\operatorname{KV}$ be one of the varieties $\operatorname{KV}_\pm$ corresponding to the chosen maximal orthogonal Grassmannian, and let $\operatorname{PK}\colon \operatorname{BQ}^\circ\to\operatorname{KV}$ be the corresponding Plücker–Klein map. We proceed to define the real Plúcker–Klein map, beginning with some general definitions. By definition, a real structure on a complex (algebraic) variety $X$ is an antiholomorphic (antiregular) involution $c\colon X\to X$. The pair $(X,c)$ is called a real (algebraic) variety and the fixed point set of the involution $c\colon X\to X$ is called the real part of this variety. The real part of $(X,c)$ is usually denoted by $\mathbb{R}X$. When the real structure $c\colon X\to X$ on a variety $X$ is clear from the context, we briefly denote the pair $(X,c)$ just by $X$. The notation for the real part of $X$ can also be abbreviated to $\mathcal{X}$. A holomorphic (regular) map $f\colon (X,c)\to(Y,c)$ between real varieties is said to be real if the diagram commutes. We also fix an antilinear involution $c\colon V\to V$ on the complex vector space $V$ and call it a real structure on $V$. The fixed point set of this involution is denoted by $\mathbb{R}V$. The form $\mathfrak{q}$ is assumed to be real, that is, $c^*(\mathfrak{q})=\mathfrak{q}$. Moreover, we require that the signature of the form $\mathfrak{q}(\mathbf{v})|_{\mathbb{R}V}$ be equal to zero. It will be explained in § 2 that this condition is necessary for the existence of real $g$-planes on $\mathfrak{Q}$.The involution $c\colon V\to V$ induces a real structure $c\colon \operatorname{BQ}\to\operatorname{BQ}$ on the space of biquadrics. Fixed points of this involution are called real biquadrics. We denote the real part of $\operatorname{BQ}$ by $\mathcal{BQ}$. The intersection $\mathcal{BQ}\cap\operatorname{BQ}^\circ$ is denoted by $\mathcal{BQ}^\circ$. The differentiable manifold $\mathcal{BQ}^\circ$ consists of smooth real biquadrics. If $B\in\mathcal{BQ}^\circ$, then the line $L(B)\subset\operatorname{Qu}$ is real, that is, $c(L(B))=L(B)$. Let $\nu(B)$ be the number of real points in the set $\{p_1,\dots,p_n\}\subset L(B)$. Since $n$ is even, so is $\nu(B)$. For every $r\in\{0,1,\dots,g+1\}$ let $\mathcal{BQ}^{(r)}$ be the set of all biquadrics $B\in\mathcal{BQ}^\circ$ with $\nu(B)=2r$. Since the function $\nu(B)$ is continuous, the set $\mathcal{BQ}^{(r)}$ consists of whole connected components of $\mathcal{BQ}^\circ$. The involution $c\colon \mathbb{P}(V)\to\mathbb{P}(V)$ induces an antiholomorphic involution $c\colon \mathfrak{Q}\,{\to}\,\mathfrak{Q}$ since the quadratic form $\mathfrak{q}$ is real. The involution $c\colon\mathfrak{Q}\,{\to}\,\mathfrak{Q}$ induces an involution $c\colon \operatorname{G}\to\operatorname{G}$ since the signature of the quadratic form $\mathfrak{q}|_{\mathbb{R}V}$ is equal to zero (see § 2). For every real biquadric $B\in\operatorname{BQ}^\circ$, the Kummer variety $K(B)\subset\operatorname{G}$ is invariant under the involution $c\colon \operatorname{G}\to\operatorname{G}$. Therefore, for every biquadric $B\in\mathcal{BQ}^\circ$, the Kummer variety $K(B)$ is a real algebraic variety. Writing $\mathcal{KV}$ for the real part of $\operatorname{KV}$, we obtain a map
$$
\begin{equation*}
\mathcal{PK}\colon \mathcal{BQ}^\circ\to\mathcal{KV}.
\end{equation*}
\notag
$$
Note that this map is not surjective (see the remark at the end of § 2). We denote its image by $\mathcal{KV}^\circ$. Let $\mathcal{KV}^{(r)}$ be the image of the differentiable manifold $\mathcal{BQ}^{(r)}$ under the Plücker–Klein map $\operatorname{PK}\colon \operatorname{BQ}^\circ\to\operatorname{KV}$. Then $\mathcal{KV}^{(r)}$ is an $(N-1)$-dimensional differentiable manifold consisting of whole connected components of $\mathcal{KV}^\circ$ and there is a surjective Plücker–Klein map
$$
\begin{equation*}
\mathcal{PK}\colon \mathcal{BQ}^{(r)}\to\mathcal{KV}^{(r)}.
\end{equation*}
\notag
$$
We write $S(2r)$ for a circle with $2r$ points removed. In particular, $S(0)$ is a circle. We shall first prove Proposition 0.1. Proposition 0.1. The map $\mathcal{PK}\colon \mathcal{BQ}^{(r)}\to\mathcal{KV}^{(r)}$ is a locally trivial bundle with fibre $S(2r)$. The manifold $\mathcal{BQ}^\circ$ is disconnected. Its connected components are called deformation classes of real biquadrics. Given any topological space $X$, we shall write $\langle X\rangle$ for the set of its connected components. The first problem solved in the present paper is to describe the sets $\langle\mathcal{BQ}^{(r)}\rangle$, $\langle\mathcal{KV}^{(r)}\rangle$ and the maps
$$
\begin{equation*}
\langle\mathcal{PK}\rangle\colon \langle\mathcal{BQ}^{(r)}\rangle \to\langle\mathcal{KV}^{(r)}\rangle.
\end{equation*}
\notag
$$
We state the results obtained in the course of solution of this problem. The set $\mathcal{BQ}^{(r)}$ consists of whole connected components of $\mathcal{BQ}^\circ$. We shall prove Proposition 0.2 (see § 3 for details). Note that Corollary 0.3 follows from Propositions 0.1, 0.2. To describe the sets $\langle\mathcal{BQ}^{(r)}\rangle$, $\langle\mathcal{KV}^{(r)}\rangle$ for $r>0$ and the corresponding map
$$
\begin{equation*}
\langle\mathcal{PQ}\rangle\colon \langle\mathcal{BQ}^{(r)}\rangle\to \langle\mathcal{KV}^{(r)}\rangle,
\end{equation*}
\notag
$$
we introduce further notation. We write $\mathcal{R}$ for the set $\{1,2,\dots,2r\}$ of positive integers and consider the set of maps from $\mathcal{R}$ to a fixed set $X$. This set of maps is denoted by $\operatorname{Map}(\mathcal{R},X)$. Let $\mathfrak{S}_{2r}$ be the group of permutations of the elements of $\mathcal{R}$. We define an action of $\mathfrak{S}_{2r}$ on $\operatorname{Map}(\mathcal{R},X)$ by the following rule. If $\sigma\in\mathfrak{S}_{2r}$, then the action of $\sigma$ on $f\in\operatorname{Map}(\mathcal{R},X)$ is given by the composition This is a right action, and we denote its result $f\circ\sigma$ by $f^\sigma$.Let $\mathcal{E}^{(r)}$ be the set of vectors in the space $(\mathbb{F}_2)^{2r}$ such that the number of zero coordinates of $\mathbf{e}$ is equal to the number of coordinates 1, that is, to $r$. This is a subset of $\operatorname{Map}(\mathcal{R},\mathbb{F}_2)$. It is invariant under the action of $\mathfrak{S}_{2r}$. Note that
$$
\begin{equation*}
(e_1,\dots,e_{2r})^\sigma=(e_{\sigma(1)},\dots,e_{\sigma(2r)}).
\end{equation*}
\notag
$$
Let $\Sigma(2r)$ be the subgroup of $\mathfrak{S}_{2r}$ of order 2 generated by the permutation $i\mapsto 2r-i+1$, and let $\Delta(2r)$ be the subgroup of $\mathfrak{S}_{2r}$ generated by the permutations $\sigma_{2r}$, $\delta_{2r}$, where $\delta_{2r}$ is the maximal cycle $(12\dots 2r)$. Theorem 0.4. For $r>0$ there are canonical bijections such that the following diagram commutes: where the map $\pi$ is induced by the inclusion $\Sigma(2r)\subset\Delta(2r)$. We shall use the following notation for the set of orbits of a group $G$ acting on a set $M$. The set of orbits of a left (resp, right) action of $G$ on $M$ is denoted by $G\setminus M$ (resp. $M/G$). The orbit is denoted by $|e_1,\dots,e_{2r}|$, and the orbit is denoted by $[e_1,\dots,e_{2r}]$. These orbits are referred to as $\mathbb{F}_2$-codes. Two connected components of $\mathcal{BQ}^\circ$ are said to be cosingular if the Plücker–Klein map sends them to the same component of $\mathcal{KV}^\circ$. Denoting the connected components of $\mathcal{BQ}^{(r)}$ and $\mathcal{KV}^{(r)}$ by the corresponding $\mathbb{F}_2$-codes, we have Corollary 0.5 of Theorem 0.4.Corollary 0.5. The following components and only these components are cosingular to the component $|e_1,\dots,e_{2r}|$: Note that some components in the list (0.2) may coincide. For example, the component $|0,1,0,1,\dots,0,1|$ has no other cosingular components. When $B$ belongs to $\mathcal{BQ}^{(0)}$, we use the empty brackets $|\ |$ for the $\mathbb{F}_2$-code of $B$. The $\mathbb{F}_2$-code of the Kummer variety $K\in\mathcal{KV}^{(0)}$ is accordingly denoted by $[\ ]$. The statement of Theorem 0.4 has the following drawback. The structure of the occurring canonical bijections is not described. The corresponding construction is given in § 4. It uses the method of Rohn [5] to associate a permutation with every real quadratic complex (see § 4.1 for details). Our proof of Theorem 0.4 also uses some arguments of Rohn [5], which had to be sharpened and complemented. Complementing the statement of Theorem 0.4 by the description of the occurring bijections in terms of the $\operatorname{mod}2$-reduction of Rohn’s coding, we obtain a result that may be called Rohn’s theorem. Note that the description of connected components of the variety of unmarked biquadrics in more recent literature uses the index function on an oriented circle (details can be found in [16], [17], [18], Theorem A4.2.2). Since Theorem A4.2.2 in [18] has an analogue for marked biquadrics, one can state Theorem 0.4 in terms of the index coding. In the present paper we deduce this analogue from Rohn’s theorem (see Corollary 5.8) although it can certainly be proved independently. However, while Rohn’s code of a component is multiplied by a cyclic permutation when we pass to a cosingular component, the change of the index function is much more complicated (see the remark after the statement of Proposition 4.3). This makes the description of the Plücker–Klein map in terms of the index coding of connected components very complicated. Therefore we use Rohn’s coding of connected components in order to completely state and prove Theorem 0.4. We actually reduce Rohn’s codes (permutations) $\operatorname{mod} 2$ to obtain sequences of zeros and ones which are $\mathbb{F}_2$-codes of connected components and are called reduced Rohn’s codes. Note that there is a simple rule for obtaining the $\mathbb{F}_2$-code of a component from its index code (see Proposition 4.3 and the subsequent remark). Therefore the description of bijections in Theorem 0.4 may be stated in terms of the index function, but the proof of commutativity of the diagram requires Rohn’s coding. This is one of the reasons for considering Rohn’s coding before index coding. Another reason is the quest for historical justice since Rohn’s coding was published about 100 years before the index coding. Finally, a mistake in [8] which I made while stating the analogue of Theorem A4.2.2 in [18] (see § 5.4 for details) forced me to use Rohn’s coding first. Note that the form $\mathfrak{q}(\mathbf{v})$ is determined by the quadric $\mathfrak{Q}$ up to multiplication by a real number $\lambda\neq0$. If we replace $\mathfrak{q}(\mathbf{v})$ by $\lambda\mathfrak{q}(\mathbf{v})$ with $\lambda>0$, then the bijection
$$
\begin{equation*}
\langle\mathcal{BQ}^{(r)}\rangle=\mathcal{E}^{(r)}/\Sigma(2r)
\end{equation*}
\notag
$$
remains unchanged. But if we replace $\mathfrak{q}(\mathbf{v})$ by $\lambda\mathfrak{q}(\mathbf{v})$ with $\lambda<0$, this bijection is changed in the following way. Replace zeros by ones and vice versa in the $\mathbb{F}_2$-code We proceed to state the next theorem. To do this, we need further notation and notions. If $B\,{\in}\,\mathcal{BQ}^\circ$, then the line $L(B)$ is real. Let $\pi\colon W(B)\to L(W)$ be the double covering of $L(B)$ branched at the points $p_1,\dots,p_n{\in}\, L(B)$. Then $W=W(B)$ is a hyperelliptic curve of genus $g$, whose real structure is uniquely determined by requiring that the projection $\pi$ is a real map and the preimage of the marked point $\mathfrak{p}$ under $\pi\colon W(B)\to L(B)$ consists of two real points. Hence the Kummer variety of the Jacobian of $W(B)$ has a canonical real structure. We denote the corresponding real algebraic variety by $K(W)$ and call it the Kummerian of $W$. Note that if the real structure of $W(B)$ is such that the preimage of $\mathfrak{p}$ has no real points, then the real structure of the Jacobian is changed by the inversion involution $(-1)\colon J(W)\to J(W)$, but the real structure of the Kummerian $K(W)$ does not change. The varieties $K(B)$, $K(W)$ are isomorphic as complex varieties but need not be isomorphic as real varieties. We shall prove Theorem 0.6. Theorem 0.6. When $B\,{\in}\,\mathcal{BQ}^{(0)}$, the real varieties $K(B)$, $K(W)$ are isomorphic if and only if $g$ is even. When $B\in\mathcal{BQ}^{(r)}$, $r>0$, the real varieties $K(B)$, $K(W)$ are isomorphic if and only if $B$ belongs to the component $|0,1,0,1,\dots,0,1|$. In what follows we also write $|0,1|^{(r)}$ for the component $|0,1,0,1,\dots,0,1|$, where the number of pairs $\{0,1\}$ is equal to $r>0$. This completes the detailed statement of our results, but we warn the reader that some results are not stated in the Introduction to avoid overload. We say only a few general words about these results. Proposition 4.3 gives a deformation classification of real marked biquadrics in terms of the index function. This classification is analogous to the classification of real unmarked biquadrics in [18], Appendix A. Besides the deformation classification of the real Plücker–Klein map, we give a rough projective classification of this map. Note that the notion of rough projective classification of real projective varieties was introduced in [19]. We give the corresponding definition for real marked biquadrics. Some elements of the group of real automorphisms of $\mathfrak{Q}$ permute the components of $\mathcal{BQ}^\circ$. We call two components of $\mathcal{BQ}^\circ$ equivalent if there is an element of the group of real automorphisms of $\mathfrak{Q}$ sending one component to the other. This is an equivalence relation on $\langle\mathcal{BQ}^\circ\rangle$. The quotient with respect to it is denoted by $\langle\!\langle\mathcal{BQ}^\circ\rangle\!\rangle$ and the cosets are called rough projective classes of real marked biquadrics. Our Theorem 5.4 and Corollary 5.6 provide an analogue of Theorem 0.4 where the deformation classification is replaced by the rough projective one. We finally note that Theorem 0.6 is complemented by Theorem 10.5 which gives a topological classification of the real parts of Kummer varieties arising from real biquadrics. We say a few words about the structure and content of the paper. Besides the Introduction, it contains ten sections. § 1 is devoted to the complex Plücker–Klein map. Here we present the results of [13] which are needed in the two next sections. In § 2 we consider the real Plücker–Klein map and state the definitions and assertions needed in the proofs of Propositions 0.1, 0.2 and Theorem 0.4. We prove Propositions 0.1, 0.2 (resp. Theorem 0.4) in § 3 (resp. § 4). § 5 is devoted to rough projective classification of real biquadrics, associated Kummer varieties, and the Plücker–Klein map. In § 6 we present further information from [13] needed in the proof of Theorem 0.6. In § 7 we state known facts about real tori and Kummer varieties to be used in what follows. In § 8 we state known facts about the Jacobians and Kummerians of real hyperelliptic curves to be used in what follows. § 9 contains a proof of Theorem 0.6. Finally, § 10 is devoted to the topological classification of Kummer varieties arising from real biquadrics. We prove Theorem 10.5 which shows how the topological type of the real part of $\mathcal{PK}(B)$ depends on the deformation class of the biquadric $B\in\mathcal{BQ}^\circ$. § 1. The complex Plücker–Klein mapIn this section we present information (see [13] for details) needed in the two next sections, which are devoted to the real Plücker–Klein map. 1.1. Fibres of the Plücker–Klein mapWe write for an arbitrary system of coordinates on $V$ which is orthonormal with respect to $\mathfrak{q}(\mathbf{v})$. Let $B=\mathfrak{Q}\cap Q$ be a biquadric and let $\operatorname{sk}=(v_1,\dots,v_n)$ be a system of coordinates. Then $B$ is given by a system of equations
$$
\begin{equation*}
v_1^2+\dots+v_n^2=0,\qquad \sum_{i,j=1}^na_{ij}v_iv_j=0,
\end{equation*}
\notag
$$
where the first (resp. second) equation determines the quadric $\mathfrak{Q}$ (resp. $Q$). In what follows we determine a biquadric in the coordinate system $\operatorname{sk}$ by giving only the second equation. If $B\in\operatorname{BQ}^\circ$, then the Weierstrass theorem on two quadratic forms yields a system of coordinates $\operatorname{sk}=(v_1,\dots,v_n)$ such that $B$ is given by the equation where the coefficients $a_1,\dots,a_n$ are pairwise distinct complex numbers. The system of coordinates $(v_1,\dots,v_n)$ is determined by the biquadric $B\in\operatorname{BQ}^\circ$ uniquely up to permuting the coordinates and multiplying some of them by $-1$. If we fix such a system of coordinates, then the coefficients $a_1,\dots,a_n$ in the equation (1.1) are determined by the biquadric $B$ uniquely up to an affine transformation
$$
\begin{equation*}
a\mapsto\alpha a+\beta,\qquad\alpha,\beta\in\mathbb{C},\quad\alpha\neq0,
\end{equation*}
\notag
$$
of the complex line $\mathbb{C}$. We fix a system of coordinates $\operatorname{sk}=(v_1,\dots,v_n)$ and write $\operatorname{BQ}(\operatorname{sk})$ for the set of biquadrics given by the equations of the form (1.1), where $a_1,\dots,a_n$ are arbitrary numbers not all equal to each other. The set $\operatorname{BQ}(\operatorname{sk})$ is a projective space of dimension $n-2$. Then we use an isomorphism $\operatorname{BQ}(\operatorname{sk})=\mathbb{P}^{n-2}$, which sends every biquadric given by an equation (1.1) to the point Let $\operatorname{BQ}^\circ(\operatorname{sk})$ be the set of all non-singular biquadrics in $\operatorname{BQ}(\operatorname{sk})$. It is the complement of the hyperplanes $\{a_i-a_j=0\}$, $0\leqslant i<j\leqslant n$. Therefore $\operatorname{BQ}^\circ(\operatorname{sk})$ is an affine algebraic variety. We write $\operatorname{KV}(\operatorname{sk})$ for the image of $\operatorname{BQ}^\circ(\operatorname{sk})$ in $\operatorname{KV}$ under the Plücker–Klein map ($\operatorname{PK}$). Thus we have a map
$$
\begin{equation*}
\operatorname{PK}\colon \operatorname{BQ}^\circ(\operatorname{sk}) \to\operatorname{KV}(\operatorname{sk}).
\end{equation*}
\notag
$$
Here is a beautiful geometric interpretation of its fibres (see [13] for details). Let $B_1,\dots,B_n\in\operatorname{BQ}(\operatorname{sk})$ be the singular biquadrics given by the equations respectively. If $B\in\operatorname{BQ}^\circ(\operatorname{sk})$, then the points $B, B_1,\dots, B_n\in\operatorname{BQ}(\operatorname{sk})=\mathbb{P}^{n-2}$ are in general position. Hence there is a unique normal rational curve
$$
\begin{equation*}
C(B)\subset\operatorname{BQ}(\operatorname{sk})=\mathbb{P}^{n-2}
\end{equation*}
\notag
$$
of degree $n-2$ passing through the points $B,B_1,\dots,B_n$. It is called a Veronese curve in [20]. It was proved in [13] that
$$
\begin{equation*}
\operatorname{PK}^{-1}(\operatorname{PK}(B))=C(B)\setminus\{B_1,\dots,B_n\}.
\end{equation*}
\notag
$$
In what follows the curve $C(B)\setminus\{B_1,\dots,B_n\}\subset\operatorname{BQ}^\circ(\operatorname{sk})$ is denoted by $C^\circ(B)$ and is called an incomplete Veronese curve. We recall that $\mathrm{M}_{0,n}$ denotes the moduli space of enumerated $n$-tuples of distinct points on the projective line. The projective equivalence class of an $n$-tuple $p_1,\dots,p_n\in\mathbb{P}^1$ is denoted by $[p_1,\dots,p_n]$. We accordingly write $\mathrm{M}_{n}$ for the moduli space of enumerated $n$-tuples of distinct points on the affine line. The affine equivalence class of an $n$-tuple $a_1,\dots,a_n\in\mathbb{C}$ is denoted by $|a_1,\dots,a_n|$. In what follows we regard $\mathrm{M}_{n}$ as an open subvariety of the projective space $\mathbb{P}^{n-2}$, where the embedding $\mathrm{M}_{n}\hookrightarrow\mathbb{P}^{n-2}$ is given by the rule For every biquadric $B\in\operatorname{BQ}^\circ(\operatorname{sk})$ with equation (1.1), we have a point Thus we obtain an isomorphism of varieties
$$
\begin{equation*}
\mathrm{J}\colon \operatorname{BQ}^\circ(\operatorname{sk}) \xrightarrow{\approx}\mathrm{M}_{n}.
\end{equation*}
\notag
$$
This isomorphism is compatible with the embeddings
$$
\begin{equation*}
\operatorname{BQ}^\circ(\operatorname{sk})\subset\mathbb{P}^{n-2}, \qquad\mathrm{M}_{n}\subset\mathbb{P}^{n-2},
\end{equation*}
\notag
$$
that is, we have a commutative diagram It follows from the description of fibres of the Plücker–Klein map that the isomorphism
$$
\begin{equation*}
\mathrm{J}\colon \operatorname{BQ}^\circ(\operatorname{sk}) \xrightarrow{\approx}\mathrm{M}_{n}
\end{equation*}
\notag
$$
induces a bijection
$$
\begin{equation*}
[\mathrm{J}]\colon \operatorname{KV}(\operatorname{sk}) \xrightarrow{\approx}\mathrm{M}_{0,n}.
\end{equation*}
\notag
$$
This bijection endows $\operatorname{KV}(\operatorname{sk})$ with the structure of an affine algebraic variety. Moreover, the following theorem holds. Theorem 1.1. Consider the map $\pi\colon \mathrm{M}_{n}\to\mathrm{M}_{0,n}$ sending every point to the point Then the diagram commutes. The image of a point $B$ (resp. the Veronese curve $C^\circ(B)$) under the isomorphism
$$
\begin{equation*}
\mathrm{J}\colon \operatorname{BQ}^\circ(\operatorname{sk})\to\mathrm{M}_{n}
\end{equation*}
\notag
$$
is denoted by $b$ (resp. $C^\circ(b)$). Let $b_1,\dots,b_n\in\mathbb{P}^{n-2}$ be the points corresponding to the singular biquadrics $B_1,\dots,B_n$ under the isomorphism $\operatorname{BQ}(\operatorname{sk})=\mathbb{P}^{n-2}$. Then where $C(b)\subset\mathbb{P}^{n-2}$ is the complete Veronese curve passing through the points $b,b_1,\dots,b_n$. 1.2. A geometric model of the Plücker–Klein mapWe write $\operatorname{SK}$ for the variety whose points are systems of coordinates $\operatorname{sk}=(v_1,\dots,v_n)$. There is a free transitive right action of the group $\mathrm{O}(\mathfrak{q},V)$ on $\operatorname{SK}$. This action is given by the rule
$$
\begin{equation*}
\operatorname{sk}\cdot f=\operatorname{sk}{\circ}\, f,
\end{equation*}
\notag
$$
where $f\in\mathrm{O}(\mathfrak{q},V)$ and $\operatorname{sk}$ is the map $V\to\mathbb{C}^n$ given by the system of coordinates $\operatorname{sk}\in\operatorname{SK}$. Since the orthogonal group $\mathrm{O}(\mathfrak{q},V)$ consists of two connected components, the variety $\operatorname{SK}$ also consists of two connected components. The permutation group $\mathfrak{S}_n$ acts on the right on the varieties $\mathrm{M}_{0,n}$, $\mathrm{M}_{n}$, $\operatorname{SK}$:
$$
\begin{equation*}
\begin{gathered} \, [p_1,\dots,p_n]^\sigma=[p_{\sigma(1)},\dots,p_{\sigma(n)}],\qquad |a_1,\dots,a_n|^\sigma=|a_{\sigma(1)},\dots,a_{\sigma(n)}|, \\ (v_1,\dots,v_n)^\sigma=(v_{\sigma(1)},\dots,v_{\sigma(n)}). \end{gathered}
\end{equation*}
\notag
$$
The map $\pi\colon \mathrm{M}_{n}\to\mathrm{M}_{0,n}$ is equivariant under this action. Denoting the multiplicative group $\{\pm1\}$ by $\mu_2$, we consider the analogous right action of $\mathfrak{S}_n$ on the group $(\mu_2)^n$:
$$
\begin{equation*}
(\eta_1,\dots,\eta_n)^\sigma=(\eta_{\sigma(1)},\dots,\eta_{\sigma(n)}),
\end{equation*}
\notag
$$
and use it to construct a semidirect product $\mathfrak{S}_n\leftthreetimes(\mu_2)^n$ in accordance with the construction in Hall’s book [21]. Namely, the group $\mathfrak{S}_n\leftthreetimes(\mu_2)^n$ consists of pairs $(\sigma,\boldsymbol{\eta})$, where
$$
\begin{equation*}
\sigma\in\mathfrak{S}_n,\qquad\boldsymbol{\eta}=(\eta_1,\dots,\eta_n)\in(\mu_2)^n,
\end{equation*}
\notag
$$
and the group operation is given by the rule
$$
\begin{equation*}
(\sigma,\boldsymbol{\eta})\cdot(\sigma',\boldsymbol{\eta}') =(\sigma\cdot\sigma',\boldsymbol{\eta}^{\sigma'}\cdot\boldsymbol{\eta}').
\end{equation*}
\notag
$$
We denote the group $\mathfrak{S}_n\leftthreetimes(\mu_2)^n$ by $\widehat{\mathfrak{S}}_n$ and consider a right action of this group on $\operatorname{SK}$ by the rule
$$
\begin{equation*}
(v_1,\dots,v_n)^{(\sigma,\boldsymbol{\eta})}=(\eta_1v_{\sigma(1)},\dots,\eta_nv_{\sigma(n)}).
\end{equation*}
\notag
$$
There are also actions of $\widehat{\mathfrak{S}}_n$ on $\mathrm{M}_{0,n}$ and $\mathrm{M}_{n}$ by permutations only, that is,
$$
\begin{equation*}
[p_1,\dots,p_n]^{(\sigma,\boldsymbol{\eta})}=[p_1,\dots,p_n]^\sigma,\qquad |a_1,\dots,a_n|^{(\sigma,\boldsymbol{\eta})}=|a_1,\dots,a_n|^\sigma.
\end{equation*}
\notag
$$
Hence $\widehat{\mathfrak{S}}_n$ acts on the right on the product $\mathrm{M}_{n}\times\operatorname{SK}$ by means of the diagonal action. Consider the map
$$
\begin{equation*}
\mathrm{M}_{n}\times\operatorname{SK}\to\operatorname{BQ}^\circ
\end{equation*}
\notag
$$
sending every pair $(|a_1,\dots,a_n|,(v_1,\dots,v_n))$ to the biquadric given by the equation Note that two pairs
$$
\begin{equation*}
(|a_1,\dots,a_n|,(v_1,\dots,v_n)),\qquad (|a'_1,\dots,a'_n|,(v'_1,\dots,v'_n))
\end{equation*}
\notag
$$
are mapped to the same point if and only if the pair belongs to the orbit
$$
\begin{equation*}
(|a_1,\dots,a_n|,(v_1,\dots,v_n))\cdot\widehat{\mathfrak{S}}_n.
\end{equation*}
\notag
$$
Hence the regular map
$$
\begin{equation*}
\mathrm{M}_{n}\times\operatorname{SK}\to\operatorname{BQ}^\circ
\end{equation*}
\notag
$$
is an unbranched covering. Denoting the geometric quotient $(\mathrm{M}_{n}\times\operatorname{SK})/\widehat{\mathfrak{S}}_n$ by $\mathfrak{BQ}$, we obtain an isomorphism of varieties
$$
\begin{equation*}
\mathfrak{I}\colon \mathfrak{BQ}\xrightarrow{\approx}\operatorname{BQ}^\circ.
\end{equation*}
\notag
$$
We also write $\mathfrak{KV}$ for the geometric quotient $(\mathrm{M}_{0,n}\times\operatorname{SK})/\widehat{\mathfrak{S}}_n$. The map
$$
\begin{equation*}
\pi\times \mathrm{id}\colon (\mathrm{M}_{n}\times\operatorname{SK})/\widehat{\mathfrak{S}}_n \to(\mathrm{M}_{0,n}\times\operatorname{SK})/\widehat{\mathfrak{S}}_n
\end{equation*}
\notag
$$
of geometric quotients is denoted by The following assertion holds (see [13]). If coordinate systems $\operatorname{sk}, \operatorname{sk}'\in\operatorname{SK}$ do not belong to the same orbit of $\widehat{\mathfrak{S}}_n$, then the varieties $\operatorname{KV}(\operatorname{sk})$ and $\operatorname{KV}(\operatorname{sk}')$ are disjoint. Hence the commutative diagram (1.2) yields the existence and uniqueness of a bijection $\mathfrak{KV}=\operatorname{KV}$ such that the diagram commutes. The map $\mathfrak{pr}$ is a submersion, and so is $\operatorname{PK}$ if we endow $\operatorname{KV}$ with the structure of an affine algebraic variety induced by the bijection $\operatorname{KV}=\mathfrak{KV}$ in (1.3).
§ 2. The real Plücker–Klein map2.1. The real marked quadratic formThe indices of inertia $I_+$, $I_-$ of the restriction of the real form $\mathfrak{q}(\mathbf{v})$ to the real vector space $\mathbb{R}V$ are well defined since the form $\mathfrak{q}(\mathbf{v})|_{\mathbb{R}V}$ is real-valued. Note that the complete Fano variety $\Phi$ of the corresponding quadric $\mathfrak{Q}$ is invariant under the involution $c\colon \mathfrak{Q}\to\mathfrak{Q}$. Hence $\Phi$ is a real variety. On the other hand, the corresponding maximal Grassmannians $\operatorname{G}_\pm$ are not always real varieties since the involution $c\colon \Phi\to\Phi$ may interchange $\operatorname{G}_+$ and $\operatorname{G}_-$. Proposition 2.1. The involution $c\colon \Phi\to\Phi$ does not interchange $\operatorname{G}_+$ and $\operatorname{G}_-$ if and only if where $I_\pm$ are the indices of inertia of $\mathfrak{q}(\mathbf{v})|_{\mathbb{R}V}$. Proof. Note that the signature $I_+-I_-$ is even and we may assume that $I_+-I_-=2k\geqslant0$. Hence there are real coordinates $x_1,\dots,x_n$ on $V$ such that
$$
\begin{equation*}
\mathfrak{q}(\mathbf{v})=x_1^2+\dots+x_{2k}^2+x_{2k+1}^2-x_{2k+2}^2+\dots+x_{n-1}^2-x_n^2.
\end{equation*}
\notag
$$
We put
$$
\begin{equation*}
\begin{gathered} \, y_1=x_1-ix_2,\quad y_2=x_1+ix_2,\quad \dots, \\ \quad y_{2k-1}=x_{2k-1}-ix_{2k},\quad y_{2k}=x_{2k-1}+ix_{2k}, \end{gathered}
\end{equation*}
\notag
$$
where $i=\sqrt{-1}$. Putting also
$$
\begin{equation*}
\begin{gathered} \, y_{2k+1}=x_{2k+1}-x_{2k+2},\quad y_{2k+2}=x_{2k+1}+x_{2k+2},\quad \dots, \\ \quad y_{n-1}=x_{n-1}-x_n,\quad y_n=x_{n-1}+x_n, \end{gathered}
\end{equation*}
\notag
$$
we obtain an equality Note that the $g$-plane $\Pi=\{y_1=y_3=\dots=y_{n-1}=0\}$ is contained in $\mathfrak{Q}$ and the intersection $\Pi\cap c(\Pi)$ is given by the system of equations Hence Therefore the $g$-planes $\Pi$ and $c(\Pi)$ belong to the same family of $g$-planes on $\mathfrak{Q}$ if and only if ${k\equiv0 \operatorname{mod} 2}$ (see [14]). $\Box$ Proof. Suppose that $I_+=I_-$. The existence of a real $g$-plane on $\mathfrak{Q}$ in this case was actually established in the proof of Proposition 2.1. (If $I_+=I_-$, then the $g$-plane $\Pi$ in that proof is real.)
To prove the converse, we define an invariant $\ell(\mathfrak{Q})$ as the maximal number $k$ such that the restriction homomorphism
$$
\begin{equation*}
H^k(\mathbb{RP}(V),\mathbb{F}_2)\to H^k(\mathbb{R}\mathfrak{Q},\mathbb{F}_2)
\end{equation*}
\notag
$$
is non-zero, provided that $\mathbb{R}\mathfrak{Q}$ is non-empty. When $\mathbb{R}\mathfrak{Q}$ is empty, we put $\ell(\mathfrak{Q})=-1$ by definition. The following equality holds (see, for example, [22]): We now notice that if there is a real $g$-plane on $\mathfrak{Q}$, then In what follows we assume that $I_+=I_-$ for the fixed quadratic form $\mathfrak{q}(\mathbf{v})$. Then Proposition 2.3 holds. Proof. It follows from Proposition 2.2 that at least one of the Grassmannians $\operatorname{G}_\pm$ has real points. There is no loss of generality in assuming that $\mathbb{R}\operatorname{G}_+\neq\varnothing$. On the other hand, there are real automorphisms of $\mathfrak{Q}$ that interchange the Grassmannians $\operatorname{G}_+$ and $\operatorname{G}_-$ (see the proof of Proposition 2.1). Hence the inequality $\mathbb{R}\operatorname{G}_+\neq\varnothing$ implies that $\mathbb{R}\operatorname{G}_-\neq\varnothing$. $\Box$ 2.2. Real systems of coordinatesIn what follows we write $\mathcal{SK}^{(r)}$ for the set of all systems of coordinates
$$
\begin{equation*}
\operatorname{sk}=(v_1,\dots,v_n)\in\operatorname{SK}
\end{equation*}
\notag
$$
such that the coordinate functions $v_1,v_3,\dots,v_{2r-1}$ are real, the coordinate functions $v_2,v_4,\dots,v_{2r}$ are imaginary, and the pairs
$$
\begin{equation*}
\{v_{2r+1},v_{2r+2}\},\quad \dots,\quad \{v_{n-1},v_n\}
\end{equation*}
\notag
$$
of coordinate functions are complex-conjugate. Let $\mathcal{O}(\mathfrak{q},V)$ be the subgroup of $\mathrm{O}(\mathfrak{q},V)$ consisting of all real transformations. Then $\mathcal{O}(\mathfrak{q},V)$ acts freely and transitively on $\mathcal{SK}^{(r)}$. The restriction homomorphism
$$
\begin{equation*}
\mathcal{O}(\mathfrak{q},V)\to\mathrm{O}(\mathfrak{q}|_{\mathbb{R}V},\mathbb{R}V)
\end{equation*}
\notag
$$
is an isomorphism. Since the group $\mathrm{O}(\mathfrak{q}|_{\mathbb{R}V},\mathbb{R}V)$ consists of four connected components, we see that the variety $\mathcal{SK}^{(r)}$ also consists of four connected components. We recall a description of the connected components of $\mathrm{O}(\mathfrak{q}|_{\mathbb{R}V},\mathbb{R}V)$. Fix a decomposition $\mathbb{R}V=W_+\oplus W_-$, where $W_+$ ($W_-$) is a subspace of dimension $m=g+1$ on which the quadratic form $\mathfrak{q}|_{\mathbb{R}V}$ is positive (negative) definite. We fix an orientation on $W_+$ ($W_-$). For every $m$-dimensional subspace $W\subset\mathbb{R}V$ such that $\mathfrak{q}|_{\mathbb{R}V}$ is positive (negative) definite, this induces an orientation on $W$ by means of the orthogonal projection $W\to W_+$ ($W\to W_-$). We say that a transformation $f\in\mathrm{O}(\mathfrak{q}|_{\mathbb{R}V},\mathbb{R}V)$ preserves the orientation of $W_+$ ($W_-$) if the isomorphism $f\colon W_+\to f(W_+)$ ($f\colon W_-\to f(W_-)$) is orientation-preserving. Let $\mathrm{O}(\mathfrak{q}|_{\mathbb{R}V},\mathbb{R}V)_{++}$ ($\mathrm{O}(\mathfrak{q}|_{\mathbb{R}V},\mathbb{R}V)_{--}$) be the subset of $\mathrm{O}(\mathfrak{q}|_{\mathbb{R}V},\mathbb{R}V)$ consisting of all transformations that preserve (reverse) the orientations of the spaces $W_\pm$. Moreover, let $\mathrm{O}(\mathfrak{q}|_{\mathbb{R}V},\mathbb{R}V)_{+-}$ ($\mathrm{O}(\mathfrak{q}|_{\mathbb{R}V},\mathbb{R}V)_{-+}$) be the subset of $\mathrm{O}(\mathfrak{q}|_{\mathbb{R}V},\mathbb{R}V)$ consisting of all transformations that preserve (reverse) the orientation of $W_+$ and reverse (preserve) the orientation of $W_-$. Then the sets
$$
\begin{equation*}
\mathrm{O}(\mathfrak{q}|_{\mathbb{R}V},\mathbb{R}V)_{++},\quad \mathrm{O}(\mathfrak{q}|_{\mathbb{R}V},\mathbb{R}V)_{--},\quad \mathrm{O}(\mathfrak{q}|_{\mathbb{R}V},\mathbb{R}V)_{+-},\quad \mathrm{O}(\mathfrak{q}|_{\mathbb{R}V},\mathbb{R}V)_{-+}
\end{equation*}
\notag
$$
are the connected components of $\mathrm{O}(\mathfrak{q}|_{\mathbb{R}V},\mathbb{R}V)$. The component $\mathrm{O}(\mathfrak{q}|_{\mathbb{R}V},\mathbb{R}V)_{++}$ contains the identity transformation. Hence it is a subgroup of $\mathrm{O}(\mathfrak{q}|_{\mathbb{R}V},\mathbb{R}V)$. It follows from the general properties of Lie groups that this subgroup is normal. As a set, the quotient group
$$
\begin{equation*}
\mathrm{O}(\mathfrak{q}|_{\mathbb{R}V},\mathbb{R}V)/ \mathrm{O}(\mathfrak{q}|_{\mathbb{R}V},\mathbb{R}V)_{++}
\end{equation*}
\notag
$$
is isomorphic to the set of components $\langle\mathrm{O}(\mathfrak{q}|_{\mathbb{R}V},\mathbb{R}V)\rangle$. This group is also easily seen to be isomorphic to $(\mathbb{Z}/2\mathbb{Z})^2$. Using the group isomorphism
$$
\begin{equation*}
\mathcal{O}(\mathfrak{q},V)\xrightarrow{\approx} \mathrm{O}(\mathfrak{q}|_{\mathbb{R}V},\mathbb{R}V),
\end{equation*}
\notag
$$
we obtain the four connected components of $\mathcal{O}(\mathfrak{q},V)$. They are denoted by
$$
\begin{equation*}
\mathcal{O}(\mathfrak{q},V)_{++},\quad \mathcal{O}(\mathfrak{q},V)_{--},\quad \mathcal{O}(\mathfrak{q},V)_{+-},\quad \mathcal{O}(\mathfrak{q},V)_{-+}
\end{equation*}
\notag
$$
respectively. Fixing a system of coordinates $\operatorname{sk}_0\in\mathcal{SK}^{(r)}$, we obtain an isomorphism of varieties
$$
\begin{equation*}
\mathcal{O}(\mathfrak{q},V)\xrightarrow{\approx}\mathcal{SK}^{(r)},
\end{equation*}
\notag
$$
which provides the four connected components
$$
\begin{equation*}
\mathcal{SK}^{(r)}_{++}, \quad\mathcal{SK}^{(r)}_{--}, \quad\mathcal{SK}^{(r)}_{+-}, \quad \mathcal{SK}^{(r)}_{-+}
\end{equation*}
\notag
$$
of the variety $\mathcal{SK}^{(r)}$. We shall need some information on the properties of separate coordinates in a coordinate system that belongs to $\mathcal{SK}^{(r)}$. Consider the symmetric bilinear form on $V^\vee$ induced by the quadratic form $\mathfrak{q}(\mathbf{v})$. Given any $\alpha,\beta\in V^\vee$, we denote the value of this form at $(\alpha,\beta)\in V^\vee\times V^\vee$ by $\alpha\ast\beta$. Then for every system of coordinates $(v_1,\dots,v_n)\in\operatorname{SK}$ we have We now state some corollaries of (2.1) for systems of coordinates $(v_1,\dots,v_n)\in\mathcal{SK}^{(r)}$. Writing $\mathcal{V}$ for the real part of the space $V^\vee$, we have
$$
\begin{equation*}
v_1,v_3,\dots,v_{2r-1}\in\mathcal{V},\qquad v_2,v_4,\dots,v_{2r}\in\sqrt{-1}\,\mathcal{V}.
\end{equation*}
\notag
$$
Putting
$$
\begin{equation*}
v_2=\sqrt{-1}\,w_2,\quad v_4=\sqrt{-1}\,w_4,\quad \dots,\quad v_{2r}=\sqrt{-1}\,w_{2r},
\end{equation*}
\notag
$$
we obtain the equalities The definition of $\mathcal{SK}^{(r)}$ now yields that
$$
\begin{equation*}
\begin{gathered} \, v_{2r+1}=x_{2r+1}+\sqrt{-1}\,x_{2r+2},\quad v_{2r+2}=x_{2r+1}-\sqrt{-1}\,x_{2r+2}, \\ \dots,\quad v_{n-1}=x_{n-1}+\sqrt{-1}\,x_{n},\quad v_{n}=x_{n-1}-\sqrt{-1}\,x_{n}, \end{gathered}
\end{equation*}
\notag
$$
where $x_{2r+1},\dots,x_n\in\mathcal{V}$. It follows from (2.1) that the linear forms are mutually orthogonal to each other and we have
$$
\begin{equation*}
\begin{gathered} \, x_{2r+1}\ast x_{2r+1}=x_{2r+3}\ast x_{2r+3}=\dots=x_{n-1}\ast x_{n-1}=\frac{1}{2}, \\ x_{2r+2}\ast x_{2r+2}=x_{2r+4}\ast x_{2r+4}=\dots=x_{n}\ast x_{n}=-\frac{1}{2}. \end{gathered}
\end{equation*}
\notag
$$
These calculations yield Lemma 2.4. Lemma 2.4. Suppose that $\operatorname{sk}=(v_1,\dots,v_n)\in\mathcal{SK}^{(r)}$. Then the following assertions hold. 1) If $k$ is odd and $k<2r$, then the transformation of $V$ resulting from multiplication of the coordinate $v_k$ by $-1$ belongs to $\mathcal{O}(\mathfrak{q},V)_{-+}$. 2) If $k$ is even and $k\leqslant2r$, then the transformation of $V$ resulting from multiplication of of the coordinate $v_k$ by $-1$ belongs to $\mathcal{O}(\mathfrak{q},V)_{+-}$. 3) If $k$ is odd and $k>2r$, then the transformation of $V$ resulting from multiplication of the coordinates $v_k$, $v_{k+1}$ by $-1$ belongs to $\mathcal{O}(\mathfrak{q},V)_{--}$. 4) If $k$ is odd and $k>2r$, then the transformation of $V$ resulting from interchanging the coordinates $v_k$, $v_{k+1}$ belongs to $\mathcal{O}(\mathfrak{q},V)_{+-}$. 5) If $k$, $l$ are odd and $k,l>2r$, then the transformation of $V$ resulting from interchanging the pairs $\{v_k, v_{k+1}\}$, $\{v_{l}, v_{l+1}\}$ of complex-conjugate coordinates belongs to $\mathcal{O}(\mathfrak{q},V)_{--}$. Let $(\mu_2)^{2r,s}$ be the subgroup of $(\mu_2)^n$, $s=m-r$, consisting of the vectors
$$
\begin{equation*}
(\delta_1,\dots,\delta_{2r},\varepsilon_1,\varepsilon_1,\dots,\varepsilon_s,\varepsilon_s),
\end{equation*}
\notag
$$
where $\delta_i,\varepsilon_j\in\mu_2$. This group acts on $\mathcal{SK}^{(r)}$ by coordinate-wise multiplication. Therefore it acts on the quadruple $\langle\mathcal{SK}^{(r)}\rangle$ of connected components. Using the first three parts of Lemma 2.4, we easily obtain Proposition 2.5. Proposition 2.5. If $r>0$, then $(\mu_2)^{2r,s}$ acts transitively on $\langle\mathcal{SK}^{(r)}\rangle$. The action of $(\mu_2)^{0,m}$ on $\langle\mathcal{SK}^{(0)}\rangle$ has two orbits, each of which consists of two elements. Components $\mathcal{SK}^{(0)}_{1}$ and $\mathcal{SK}^{(0)}_{2}$ are in the same orbit if and only if the determinants of the transition matrices from the systems of coordinates in $\mathcal{SK}^{(0)}_{1}$ to those in $\mathcal{SK}^{(0)}_{2}$ are equal to one. 2.3. A geometrical model of the real Plücker–Klein mapSince the quadratic form $\mathfrak{q}(\mathbf{v})|_{\mathbb{R}V}$ is of signature zero, it is easy to verify that Proposition 2.6 holds (see, for example, [23], [24]). Proposition 2.6. If $B\in\mathcal{BQ}^{(r)}$, then there is a system of coordinates Note that the first equation in (2.2) determines the quadric $\mathfrak{Q}$. In what follows we write the second equation only. The system of coordinates in Proposition 2.6 is determined by the biquadric $B\in\mathcal{BQ}^{(r)}$ uniquely up to the following transformations: permutation of real (imaginary) coordinates, permutation of complex-conjugate coordinates, interchange of coordinates in complex-conjugate pairs, multiplication of some real (imaginary) coordinates by $-1$, and multiplication of some pairs of complex-conjugate coordinates by $-1$. If we fix the system of coordinates $\operatorname{sk}=(v_1,\dots,v_n)$ considered in Proposition 2.6 and if we know that the biquadric $B\in\mathcal{BQ}^{(r)}$ is given by a system of equations of the form (2.1) in this system of coordinates, then the vector of coefficients $(a_1,\dots,a_n)$ is determined uniquely up to a real affine transformation of the complex line $\mathbb{C}$. Let $\mathfrak{S}_{n,r}$ ($\widehat{\mathfrak{S}}_{n,r}$) be the subgroup of $\mathfrak{S}_n$ ($\widehat{\mathfrak{S}}_n$) consisting of the elements that leave $\mathcal{SK}^{(r)}$ invariant. The structure of these groups is clear from the comment after Proposition 2.6. In particular, all permutations comprising the group $\mathfrak{S}_{n,r}$ are listed there. We now give exact statements. Let $\mathfrak{S}^{\mathrm{od}}_r$ ($\mathfrak{S}^{\mathrm{ev}}_r$) be the subgroup of $\mathfrak{S}_{n}$ consisting of all permutations of odd (even) numbers in $\{1,2,\dots,2r\}$ and let $\mathfrak{S}'_s$, $s=g+1-r$, be the subgroup of $\mathfrak{S}_{n}$ consisting of all permutations of the set of pairs $\{2r+1,2r+2\}$, $\dots$, $\{n-1,n\}$. Besides the subgroups $\mathfrak{S}_r^{\mathrm{od}}$, $\mathfrak{S}_r^{\mathrm{ev}}$, $\mathfrak{S}'_s$, we also consider subgroups $\mathfrak{S}_2^{(1)}$, $\dots$, $\mathfrak{S}_2^{(s)}$ of order two in $\mathfrak{S}_n$ that are the groups of permutations of elements in the pairs $\{2r+1,2r+2\}$, $\dots$, $\{n-1,n\}$ and denote the product of these groups by $\mathfrak{S}''_s$. Then the following Proposition 2.7 can be obtained from the remarks after Proposition 2.6. Let $\operatorname{sk}=(v_1,\dots,v_n)\in\mathcal{SK}^{(r)}$ be the fixed system of coordinates. We write $\mathcal{BQ}^{(r)}(\operatorname{sk})$ for the set of biquadrics in $\mathcal{BQ}^{\circ}$ which are given by the equations where $a_1,\dots,a_{2r}$ are pairwise distinct real numbers, every pair of coefficients
$$
\begin{equation*}
\{a_{2r+1},a_{2r+2}\},\quad \dots,\quad \{a_{n-1},a_n\}
\end{equation*}
\notag
$$
is a pair of complex-conjugate numbers and all these pairs are pairwise distinct. The image of the set $\mathcal{BQ}^{(r)}(\operatorname{sk})$ under the isomorphism $\operatorname{BQ}^\circ(\operatorname{sk})=\mathrm{M}_n$ is denoted by $\mathcal{M}_n^{(r)}$. Note that the differentiable manifold $\mathcal{M}_n^{(r)}$ consists of the points where $a_1,\dots,a_{2r}$ are pairwise distinct real numbers, every pair of coefficients
$$
\begin{equation*}
\{a_{2r+1},a_{2r+2}\},\quad \dots,\quad \{a_{n-1},a_n\}
\end{equation*}
\notag
$$
is a pair of complex-conjugate numbers and all these pairs are pairwise distinct. We write $\mathcal{KV}^{(r)}(\operatorname{sk})$ for the image of $\mathcal{BQ}^{(r)}(\operatorname{sk})$ under the Plücker–Klein map. The image of $\mathcal{KV}^{(r)}(\operatorname{sk})$ under the isomorphism $\mathrm{KV}(\operatorname{sk})=\mathrm{M}_{0,n}$ is denoted by $\mathcal{M}^{(r)}_{0,n}$. It consists of the points where $a_1,\dots,a_{2r}$ are pairwise distinct real numbers, every pair of coefficients
$$
\begin{equation*}
\{a_{2r+1},a_{2r+2}\},\quad\dots,\quad\{a_{n-1},a_n\}
\end{equation*}
\notag
$$
is a pair of complex-conjugate numbers and all these pairs are pairwise distinct. Note that the arguments in § 1.2 yield a commutative diagram Remark. It follows from the definitions that We claim that the variety $\mathcal{KV}^\circ$ is not equal to $\mathcal{KV}$. To prove this, we give an example of a Kummer variety $K\in\mathcal{KV}\setminus\mathcal{KV}^\circ$. Let $\operatorname{sk}=(v_1,\dots,v_n)$ be a system of coordinates in $\mathcal{SK}^{(0)}$ and let $B\in\operatorname{BQ}^\circ\setminus\mathcal{BQ}^\circ$ be the biquadric with equation where $a_1=r_1e^{i\theta_1}$, $\dots$, $a_{m}=r_me^{i\theta_m}$ are pairwise distinct complex numbers and the bar stands for complex conjugation. We easily see that the Veronese curve $C^\circ(B)$ is invariant under the involution $c\colon \operatorname{BQ}^\circ\to\operatorname{BQ}^\circ$ but contains no real points. Therefore the Kummer variety $\operatorname{PK}(B)$ belongs to $\mathcal{KV}\setminus\mathcal{KV}^\circ$.§ 3. Proof of Propositions 0.1, 0.2Consider the fibration $\pi\colon \mathrm{M}_n\to\mathrm{M}_{0,n}$ into incomplete Veronese curves. It induces a fibration $\pi\colon \overline{\mathrm{M}}_n\to\mathrm{M}_{0,n}$ into complete Veronese curves, where $\overline{\mathrm{M}}_n$ is obtained by completing each fibre of $\mathrm{M}_n$ to a complete Veronese curve. Here is a geometric construction of this completion. Any two complete Veronese curves intersect each other transversally at the base points $b_1,\dots,b_n\in\mathbb{P}^{n-2}$ (see [20], Proposition 2.10). We blow up these points by a $\sigma$-process. Then the projective space $\mathbb{P}^{n-2}$ becomes a variety $\overline{\mathbb{P}^{n-2}}$ where the proper transforms of the complete Veronese curves are disjoint. These proper transforms sweep out the variety $\overline{\mathrm{M}}_n$. We claim that the fibration $\pi\colon \overline{\mathrm{M}}_n\to\mathrm{M}_{0,n}$ is a trivial bundle whose fibre is the projective line $\mathbb{P}^1$. Indeed, consider the intersections of the variety $\overline{\mathrm{M}}_n\subset\overline{\mathbb{P}^{n-2}}$ with the exceptional projective spaces obtained by blowing up the points $b_1,\dots,b_n\in\mathbb{P}^{n-2}$. They are sections of the bundle $\pi\colon \overline{\mathrm{M}}_n\to\mathrm{M}_{0,n}$. Therefore this bundle is trivial. We define a real structure
$$
\begin{equation*}
c_r\colon \mathrm{M}_n\to\mathrm{M}_n, \qquad r=0,1,\dots,g+1,
\end{equation*}
\notag
$$
on $\mathrm{M}_n$ by the rule
$$
\begin{equation*}
c_r(|a_1,\dots,a_n|)= |\overline{a}_1,\dots,\overline{a}_{2r}, \overline{a}_{2r+2},\overline{a}_{2r+1},\dots, \overline{a}_{n},\overline{a}_{n-1}|,
\end{equation*}
\notag
$$
where the bar stands for complex conjugation. The real part of this structure is equal to $\mathcal{M}_n^{(r)}$. The real structure $c_r\colon \mathrm{M}_n\to\mathrm{M}_n$ extends to a real structure on $\overline{\mathrm{M}}_n$. Denoting the real part of $(\overline{\mathrm{M}}_n, c_r)$ by $\overline{\mathcal{M}}_n^{\,(r)}$, we obtain locally trivial bundles $\pi\colon \overline{\mathcal{M}}_n^{\,(r)}\to\mathcal{M}_{0,n}^{(r)}$ whose fibre is a circle. This follows from the construction (given above) of the map $\pi\colon \overline{\mathrm{M}}_n\to\mathrm{M}_{0,n}$. Consider the fibration $\pi\colon \mathcal{M}_n^{(0)}\to\mathcal{M}_{0,n}^{(0)}$. It coincides with the fibration
$$
\begin{equation*}
\pi\colon \overline{\mathcal{M}}_n^{\,(0)}\to\mathcal{M}_{0,n}^{(0)}.
\end{equation*}
\notag
$$
Therefore the fibration
$$
\begin{equation*}
\pi\colon \mathcal{M}_n^{(0)}\to\mathcal{M}_{0,n}^{(0)}
\end{equation*}
\notag
$$
is a locally trivial bundle with fibre $S(0)$. Using the commutative diagram (2.3), we see that the map
$$
\begin{equation*}
\mathcal{PK}\colon \mathcal{BQ}^{(0)}\to\mathcal{KV}^{(0)}
\end{equation*}
\notag
$$
is a locally trivial bundle with fibre $S(0)$. This proves Proposition 0.1 for $r=0$. We proceed to prove it for $r>0$. Consider the fibration $\pi\colon \mathcal{M}_n^{(r)}\to\mathcal{M}_{0,n}^{(r)}$. It is obtained by deleting $2r$ sections of the bundle $\pi\colon \overline{\mathcal{M}}_n^{\,(r)}\to\mathcal{M}_{0,n}^{(r)}$, whose fibre is a circle. Since these sections are disjoint, $\pi\colon \mathcal{M}_n^{(r)}\to\mathcal{M}_{0,n}^{(r)}$ is a trivial bundle with fibre $S(2r)$. Using the commutative diagram (2.3), we obtain Proposition 0.1 for $r>0$. This completes the proof of Proposition 0.1. In the course of the proof we have established the following Lemma 3.1. Lemma 3.1. If $r>0$, then the map $\pi\colon \mathcal{M}_n^{(r)}\to\mathcal{M}_{0,n}^{(r)}$ is a trivial bundle with fibre $S(2r)$. Proceeding to prove Proposition 0.2, we first describe the connected components of $\mathcal{M}_n^{(0)}$. To do this, we define a two-valued function
$$
\begin{equation*}
\operatorname{sgn}\colon \mathbb{C}\setminus\mathbb{R}\to\mu_2
\end{equation*}
\notag
$$
by putting
$$
\begin{equation*}
\operatorname{sgn}(x+iy)= \begin{cases} 1, &y>0, \\ -1, &y<0. \end{cases}
\end{equation*}
\notag
$$
Let $\mathscr{M}_n$ be the subspace of $(\mu_2)^n$ consisting of the vectors
$$
\begin{equation*}
(\varepsilon_1,-\varepsilon_1,\dots,\varepsilon_m,-\varepsilon_m),\qquad m=g+1,
\end{equation*}
\notag
$$
where $\varepsilon_1,\dots,\varepsilon_m\in\mu_2$. We write $\mathscr{F}_n$ for the quotient set $\mathscr{M}_n/(-1)$, where the equivalence relation is given by multiplication of all coordinates by $-1$. The equivalence class of a vector $(\varepsilon_1,-\varepsilon_1,\dots,\varepsilon_m,-\varepsilon_m)$ is denoted by
$$
\begin{equation*}
|\varepsilon_1,-\varepsilon_1,\dots,\varepsilon_m,-\varepsilon_m|.
\end{equation*}
\notag
$$
If $b\in\mathcal{M}_n^{(0)}$, then we can assume that where the numbers $a_1,\dots,a_{m}\in\mathbb{C}\setminus\mathbb{R}$ are pairwise distinct. They are uniquely determined up to a simultaneous real affine transformation. Since
$$
\begin{equation*}
\operatorname{sgn}(\alpha(x+iy)+\beta)= \begin{cases} \operatorname{sgn}(x+iy), &\alpha>0, \\ -\operatorname{sgn}(x+iy), &\alpha<0, \end{cases}
\end{equation*}
\notag
$$
where $\alpha,\beta\in\mathbb{R}$, $\alpha\neq0$, we see that the element
$$
\begin{equation*}
\operatorname{sgn}(b)= |{\operatorname{sgn}(a_1),\operatorname{sgn}(\overline{a}_1),\dots,\operatorname{sgn}(a_{m}), \operatorname{sgn}(\overline{a}_m)}|\in\mathscr{F}_n
\end{equation*}
\notag
$$
is uniquely determined. The resulting map
$$
\begin{equation*}
\operatorname{sgn}\colon \mathcal{M}_n^{(0)}\to\mathscr{F}_n
\end{equation*}
\notag
$$
induces a bijection
$$
\begin{equation*}
\operatorname{sgn}\colon \langle\mathcal{M}_n^{(0)}\rangle\xrightarrow{\approx} \mathscr{F}_n.
\end{equation*}
\notag
$$
The group $\mathfrak{S}_{n,0}=\mathfrak{S}'_{m}\times\mathfrak{S}''_{m}$ acts on the right on $\mathscr{F}_n$ in the standard way:
$$
\begin{equation*}
|\varepsilon_1,\varepsilon_2,\dots,\varepsilon_{n-1},\varepsilon_n|^\sigma= |\varepsilon_{\sigma(1)},\varepsilon_{\sigma(2)},\dots,\varepsilon_{\sigma(n-1)}, \varepsilon_{\sigma(n)}|.
\end{equation*}
\notag
$$
It follows from the definitions of the quotient set $\mathscr{F}_n$ and the subgroup $\mathfrak{S}''_{m}$ of $\mathfrak{S}_{n,0}$ that the action of $\mathfrak{S}''_{m}$ is transitive on $\mathscr{F}_n$. Hence the quotient set $\mathscr{F}_n/\mathfrak{S}''_{m}$ is a singleton. The bijection
$$
\begin{equation*}
\operatorname{sgn}\colon \langle\mathcal{M}_n^{(0)}\rangle\xrightarrow{\approx} \mathscr{F}_n
\end{equation*}
\notag
$$
is equivariant under the action of $\mathfrak{S}_{n,0}$. Consider the diffeomorphism of manifolds
$$
\begin{equation*}
\mathcal{BQ}^{(0)}=(\mathcal{M}_n^{(0)}\times\mathcal{SK}^{(0)})/\widehat{\mathfrak{S}}_{n,0}
\end{equation*}
\notag
$$
in § 2.3. Then we obtain equalities
$$
\begin{equation*}
\begin{aligned} \, \langle\mathcal{BQ}^{(0)}\rangle &= \langle(\mathcal{M}_n^{(0)}\times\mathcal{SK}^{(0)})/\widehat{\mathfrak{S}}_{n,0}\rangle \\ &= (\langle\mathcal{M}_n^{(0)}\rangle\times \langle\mathcal{SK}^{(0)}\rangle)/\widehat{\mathfrak{S}}_{n,0}= (\mathscr{F}_n\times\langle\mathcal{SK}^{(0)}\rangle)/\widehat{\mathfrak{S}}_{n,0}. \end{aligned}
\end{equation*}
\notag
$$
The group $\widehat{\mathfrak{S}}_{n,0}$ contains a subgroup $\mathfrak{S}_{n,0}$, which consists of permutations, and a normal subgroup $(\mu_2)^{0,m}$ such that the quotient group $\widehat{\mathfrak{S}}_{n,0}/(\mu_2)^{0,m}$ is equal to $\mathfrak{S}_{n,0}$. Since the action of $(\mu_2)^{0,m}$ on $\langle\mathcal{M}_n^{(0)}\rangle=\mathscr{F}_n$ is trivial, we have
$$
\begin{equation}
\langle\mathcal{BQ}^{(0)}\rangle= \bigl((\mathscr{F}_n\times\langle\mathcal{SK}^{(0)}\rangle)/(\mu_2)^{0,m}\bigr)/\mathfrak{S}_{n,0}.
\end{equation}
\tag{3.1}
$$
The quotient set $\langle\mathcal{SK}^{(0)}\rangle/(\mu_2)^{0,m}$ consists of two elements (see Proposition 2.5). We denote them by $\langle\mathcal{SK}^{(0)}\rangle_1$ and $\langle\mathcal{SK}^{(0)}\rangle_2$. Since $\mathfrak{S}_{n,0}=\mathfrak{S}'_{m}\times\mathfrak{S}''_{m}$, it follows from (3.1) that
$$
\begin{equation}
\langle\mathcal{BQ}^{(0)}\rangle= \bigl((\mathscr{F}_n\times \{\langle\mathcal{SK}^{(0)}\rangle_1,\langle \mathcal{SK}^{(0)}\rangle_2\})/\mathfrak{S}''_{m}\bigr)/\mathfrak{S}'_{m}.
\end{equation}
\tag{3.2}
$$
The quotient set
$$
\begin{equation*}
(\mathscr{F}_n\times\{\langle\mathcal{SK}^{(0)}\rangle_1,\langle \mathcal{SK}^{(0)}\rangle_2\})/\mathfrak{S}''_{m}
\end{equation*}
\notag
$$
has at most two elements since $\mathfrak{S}''_{m}$ acts transitively on $\mathscr{F}_n$. It remains to note that $\mathfrak{S}'_{m}$ acts transitively on the quotient set. This can be proved by using the fourth and fifth parts of Lemma 2.4. Proposition 0.2 is proved.
§ 4. Deformation classificationIn this section we prove Theorem 0.4 and Corollary 0.5. 4.1. Rohn’s codingFirst of all, we prove the following assertion. Proof. With every point whose coordinates $a_1,\dots,a_{2r}$ are pairwise distinct, we associate a sequence of positive integers which are the numbers of the coordinates $a_1,\dots,a_{2r}$ in the reverse order of their location on the real line $\mathbb{R}$, that is, $i_1$ is the number of the largest coordinate, $i_2$ is the number of the largest coordinate in the set obtained by removing $a_{i_1}$, and so on. This correspondence is called Rohn’s coding since it was used by Rohn [5]. The sequence $(i_1,\dots,i_{2r})$ of positive integers is denoted by $\operatorname{cd}(a_1,\dots,a_{2r})$. We regard it as a permutation of the numbers $\{1,2,\dots,2r\}$. Since we have
$$
\begin{equation*}
\operatorname{cd}(\alpha a_1+\beta,\dots,\alpha a_{2r}+\beta)= \begin{cases} \operatorname{cd}(a_1,\dots,a_{2r}), &\alpha>0, \\ \operatorname{cd}(a_1,\dots,a_{2r})\cdot\sigma_{2r}, &\alpha<0, \end{cases}
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\sigma_{2r}= \begin{pmatrix} 1 & 2 & \dots & 2r-1 & 2r \\ 2r & 2r-1 & \dots & 2 & 1 \end{pmatrix},
\end{equation*}
\notag
$$
we obtain a map
$$
\begin{equation*}
\operatorname{cd}\colon \mathcal{M}_{n}^{(r)}\to\mathfrak{S}_{2r}/\Sigma(2r).
\end{equation*}
\notag
$$
Indeed, if $b\in\mathcal{M}_n^{(r)}$, then we can assume that where $a_1,\dots,a_{2r}\in\mathbb{R}$, $a_{2r+1},a_{2r+3},\dots,a_{n-1}\in\mathbb{C}\setminus\mathbb{R}$ are pairwise distinct. They are uniquely determined up to a simultaneous real affine transformation. Hence the coset is well defined.
We now describe the connected components of $\mathcal{M}_n^{(r)}$. If
$$
\begin{equation*}
b=|a_1,\dots,a_{2r},a_{2r+1},\overline{a}_{2r+1},\dots,a_{n-1},\overline{a}_{n-1}| \in\mathcal{M}_n^{(r)},
\end{equation*}
\notag
$$
then we define an extended code of $b$ as the tuple
$$
\begin{equation*}
\bigl(\operatorname{cd}(a_1,\dots,a_{2r}),\operatorname{sgn}(a_{2r+1},\overline{a}_{2r+1},\dots, a_{n-1},\overline{a}_{n-1})\bigr)\in\mathfrak{S}_{2r}\times\mathscr{M}_{2s}.
\end{equation*}
\notag
$$
This yields a map
$$
\begin{equation*}
(\operatorname{cd},\operatorname{sgn})\colon \mathcal{M}_n^{(r)}\to(\mathfrak{S}_{2r} \times\mathscr{M}_{2s})/\Sigma(n,r),
\end{equation*}
\notag
$$
where $\Sigma(n,r)$ is the subgroup of order two in $\mathfrak{S}_{2r}\times(\mu_2)^{2s}$ generated by the element $(\sigma_{2r},(-1,\dots,-1))$ and the right action of $\Sigma(n,r)$ on $\mathfrak{S}_{2r}\times\mathscr{M}_{2s}$ is given by the rule
$$
\begin{equation*}
(\sigma;\varepsilon_1,\dots,\varepsilon_{2s})\cdot(\sigma_{2r},(-1,\dots,-1))= (\sigma\cdot\sigma_{2r};-\varepsilon_1,\dots,-\varepsilon_{2s}).
\end{equation*}
\notag
$$
This map induces a bijection
$$
\begin{equation*}
(\operatorname{cd},\operatorname{sgn})\colon \langle \mathcal{M}_n^{(r)}\rangle\xrightarrow{\approx} (\mathfrak{S}_{2r}\times\mathscr{M}_{2s})/\Sigma(n,r).
\end{equation*}
\notag
$$
Consider the equality
$$
\begin{equation*}
\langle\mathcal{M}_n^{(r)}\times\mathcal{SK}^{(r)}\rangle/\widehat{\mathfrak{S}}_{n,r}= (\langle\mathcal{M}_n^{(r)}\rangle\times\langle\mathcal{SK}^{(r)}\rangle)/ \widehat{\mathfrak{S}}_{n,r}.
\end{equation*}
\notag
$$
The group $\widehat{\mathfrak{S}}_{n,r}$ contains a subgroup $\mathfrak{S}_{n,r}$, which consists of permutations, and a normal subgroup $(\mu_2)^{2r,s}$ such that the quotient group $\widehat{\mathfrak{S}}_{n,r}/(\mu_2)^{2r,s}$ is equal to $\mathfrak{S}_{n,r}$. Since $(\mu_2)^{2r,s}$ acts trivially on $\langle\mathcal{M}_n^{(r)}\rangle$ and transitively on $\langle\mathcal{SK}^{(r)}\rangle$, we have
$$
\begin{equation}
\langle\mathcal{M}_n^{(r)}\times\mathcal{SK}^{(r)}\rangle/\widehat{\mathfrak{S}}_{n,r}= \langle\mathcal{M}_n^{(r)}\rangle/\mathfrak{S}_{n,r}.
\end{equation}
\tag{4.1}
$$
The bijection
$$
\begin{equation*}
(\operatorname{cd},\operatorname{sgn})\colon \langle \mathcal{M}_n^{(r)}\rangle\xrightarrow{\approx} (\mathfrak{S}_{2r}\times\mathscr{M}_{2s})/\Sigma(n,r)
\end{equation*}
\notag
$$
transfers the right action of $\mathfrak{S}_{n,r}$ to $(\mathfrak{S}_{2r}\times\mathscr{M}_{2s})/\Sigma(n,r)$. Since this action appears to be non-standard, we denote it by $\mathfrak{c}*\sigma$, where
$$
\begin{equation*}
\mathfrak{c}\in(\mathfrak{S}_{2r}\times\mathscr{M}_{2s})/\Sigma(n,r),\qquad \sigma\in\mathfrak{S}_{n,r}.
\end{equation*}
\notag
$$
To calculate this action, we prove the following assertion. If a permutation $\sigma\in\mathfrak{S}_{n,r}$ can be written in the form $\sigma=\sigma_0\times\sigma'\times\sigma''$, where
$$
\begin{equation*}
\sigma_0\in\mathfrak{S}^{\mathrm{ev}}_r\times\mathfrak{S}^{\mathrm{od}}_r,\qquad \sigma'\in\mathfrak{S}'_s,\qquad \sigma''\in\mathfrak{S}''_s,
\end{equation*}
\notag
$$
and we are given a vector $(a_1,\dots,a_{n})$, where the coordinates $a_1,\dots,a_{2r}$ are pairwise distinct real numbers, every pair of coefficients
$$
\begin{equation*}
\{a_{2r+1},a_{2r+2}\},\quad\dots,\quad\{a_{n-1},a_n\}
\end{equation*}
\notag
$$
is a pair of complex-conjugate numbers and all these pairs are distinct, then
$$
\begin{equation}
(\operatorname{cd},\operatorname{sgn})\bigl((a_1,\dots,a_{n})^{\sigma}\bigr)= \bigl(\sigma_0^{-1}\cdot\operatorname{cd}(a_1,\dots,a_{2r}), (\operatorname{sgn}(a_{2r+1},\dots,a_{n}))^{\sigma'\times\sigma''}\bigr).
\end{equation}
\tag{4.2}
$$
Note that to prove (4.2), it suffices to establish that
$$
\begin{equation}
\operatorname{cd}\bigl((a_1,\dots,a_{2r})^{\sigma}\bigr)= \sigma^{-1}\cdot\operatorname{cd}(a_1,\dots,a_{2r}),
\end{equation}
\tag{4.3}
$$
where $\sigma\in\mathfrak{S}_{2r}$. To prove the last equality, we put
$$
\begin{equation*}
\operatorname{cd}(a_1,\dots,a_{2r})=(i_1,\dots,i_{2r}),\qquad \operatorname{cd}\bigl((a_1,\dots,a_{2r})^{\sigma}\bigr)=(j_1,\dots,j_{2r}).
\end{equation*}
\notag
$$
It follows from the definition of Rohn’s $\mathfrak{S}_{2r}$-coding that $i_1$ is the number of the largest term in the sequence $a_1,\dots,a_{2r}$ and $j_1$ is the number of the largest term in the sequence $b_1=a_{\sigma(1)}$, $\dots$, $b_{2r}=a_{\sigma(2r)}$. Hence $i_1=\sigma(j_1)$, that is, $j_1=\sigma^{-1}(i_1)$. The equalities $j_k=\sigma^{-1}(i_k)$ for $k>1$ can be proved in a similar way. This proves (4.3) and, therefore, (4.2).
We now construct a right action of the group $\mathfrak{S}_{n,r}$ on the set $\mathfrak{S}_{2r}\times\mathscr{M}_{2s}$ by the following rule. If a permutation $\sigma\in\mathfrak{S}_{n,r}$ can be decomposed in the form $\sigma=\sigma_0\times\sigma'\times\sigma''$, where
$$
\begin{equation*}
\sigma_0\in\mathfrak{S}^{\mathrm{ev}}_r\times\mathfrak{S}^{\mathrm{od}}_r,\qquad \sigma'\in\mathfrak{S}'_s,\qquad \sigma''\in\mathfrak{S}''_s,
\end{equation*}
\notag
$$
and we are given an element $\mathfrak{c}=(\eta,\varepsilon)$, where $\eta\in\mathfrak{S}_{2r}$, $\varepsilon\in\mathscr{M}_{2s}$, then we define
$$
\begin{equation*}
\mathfrak{c}*\sigma=(\sigma^{-1}_0\cdot\eta,\varepsilon^{\sigma'\times\sigma''}).
\end{equation*}
\notag
$$
The resulting right action of $\mathfrak{S}_{n,r}$ on $\mathfrak{S}_{2r}\times\mathscr{M}_{2s}$ commutes with the right action of $\Sigma(n,r)$. Therefore the action of $\mathfrak{S}_{n,r}$ on $\mathfrak{S}_{2r}\times\mathscr{M}_{2s}$ descends to a right action on the set of orbits $(\mathfrak{S}_{2r}\times\mathscr{M}_{2s})/\Sigma(n,r)$. Note that if we endow the set $(\mathfrak{S}_{2r}\times\mathscr{M}_{2s})/\Sigma(n,r)$ with this right action of $\mathfrak{S}_{n,r}$, then it follows from (4.2) that the bijection
$$
\begin{equation*}
(\operatorname{cd},\operatorname{sgn})\colon \langle \mathcal{M}_n^{(r)}\rangle\xrightarrow{\approx} (\mathfrak{S}_{2r}\times\mathscr{M}_{2s})/\Sigma(n,r)
\end{equation*}
\notag
$$
is equivariant under the action of $\mathfrak{S}_{n,r}$.
Using (4.1), we obtain a bijection
$$
\begin{equation*}
\langle\mathcal{M}_n^{(r)}\times\mathcal{SK}^{(r)}\rangle/\widehat{\mathfrak{S}}_{n,r} \xrightarrow{\approx}\bigl((\mathfrak{S}_{2r}\times\mathscr{M}_{2s})/\Sigma(n,r) \bigr)/\mathfrak{S}_{n,r}.
\end{equation*}
\notag
$$
It remains to notice that since $\mathfrak{S}'_s\times\mathfrak{S}''_s$ acts transitively on $\mathscr{M}_{2s}$ and the right action of $\mathfrak{S}_r^{\mathrm{od}}\times\mathfrak{S}_r^{\mathrm{ev}}$ on $\mathfrak{S}_{2r}\times\mathscr{M}_{2s}$ can be represented in terms of a left action, namely,The following Lemma 4.2 can be proved in a similar way. Lemma 4.2. If $r>0$, then there is a canonical bijection Proof. Let $\Delta(n,r)$ be the subgroup of $\mathfrak{S}_{2r}\times(\mu_2)^{2s}$ generated by the elements
$$
\begin{equation*}
(\sigma_{2r},(-1,\dots,-1)),\qquad(\delta_{2r},(1,\dots,1)).
\end{equation*}
\notag
$$
The group $\Delta(n,r)$ will be used in our description of the set $\langle\mathcal{M}_{0,n}^{(r)}\rangle$. We recall that the fibre $\pi^{-1}(\pi(b))$ of the map $\pi\colon \mathcal{M}_n^{(r)}\to\mathcal{M}_{0,n}^{(r)}$ over a point
$$
\begin{equation*}
b=|a_1,\dots,a_{2r},a_{2r+1},\overline{a}_{2r+1},\dots,a_{n-1}, \overline{a}_{n-1}|\in\mathcal{M}_n^{(r)}
\end{equation*}
\notag
$$
contains only the following points, except for $b$:
$$
\begin{equation*}
b(t)=\biggl|\frac{1}{t-a_1},\dots,\frac{1}{t-a_{2r}},\frac{1}{t-a_{2r+1}}, \frac{1}{t-\overline{a}_{2r+1}}, \dots,\frac{1}{t-a_{n-1}},\frac{1}{t-\overline{a}_{n-1}}\biggr|,
\end{equation*}
\notag
$$
where $t\in\mathbb{R}\setminus\{a_1,\dots,a_{2r}\}$. If the extended code of $b$ is
$$
\begin{equation*}
\bigl((i_1,\dots,i_{2r}), (\varepsilon_1,-\varepsilon_1,\dots,\varepsilon_s,-\varepsilon_s)\bigr)\, \operatorname{mod} \Sigma(n,r),
\end{equation*}
\notag
$$
then the extended codes of the other points in $\pi^{-1}(\pi(b))$ are of the form
$$
\begin{equation*}
\bigl((i_1,\dots,i_{2r})\cdot\delta_{2r}^k, (\varepsilon_1,-\varepsilon_1,\dots,\varepsilon_s,-\varepsilon_s)\bigr)\, \operatorname{mod} \Sigma(n,r),
\end{equation*}
\notag
$$
where $k=1,\dots,2r-1$. Therefore the coset
$$
\begin{equation*}
\bigl((i_1,\dots,i_{2r}), (\varepsilon_1,-\varepsilon_1,\dots, \varepsilon_s,-\varepsilon_s)\bigr)\, \operatorname{mod} \Delta(n,r)
\end{equation*}
\notag
$$
should be taken for the extended code of $\pi(b)\in\mathcal{M}_{0,n}^{(r)}$ and we obtain a map
$$
\begin{equation*}
(\operatorname{cd},\operatorname{sgn})\colon \mathcal{M}_{0,n}^{(r)}\to (\mathfrak{S}_{2r}\times\mathscr{M}_{2s})/\Delta(n,r).
\end{equation*}
\notag
$$
It follows from Lemma 3.1 that this map induces a bijection
$$
\begin{equation*}
(\operatorname{cd},\operatorname{sgn})\colon \langle\mathcal{M}_{0,n}^{(r)}\rangle \xrightarrow{\approx} (\mathfrak{S}_{2r}\times\mathscr{M}_{2s})/\Delta(n,r)
\end{equation*}
\notag
$$
and we also obtain a commutative diagram This diagram yields the following commutative diagram: We also have a commutative diagram Combining these diagrams, we obtain the commutative diagram (4.4). $\Box$ Using the commutative diagram (2.3), we obtain a commutative diagram4.2. Reduced Rohn’s codingTo complete the proof of Theorem 0.4, we consider the map $\mathfrak{S}_{2r}\to(\mathbb{F}_2)^{2r}$ that sends every permutation $(i_1,\dots,i_{2r})$ to the vector
$$
\begin{equation*}
(i_1\, \operatorname{mod} 2,\ \dots,\ i_{2r}\, \operatorname{mod}2).
\end{equation*}
\notag
$$
This map has image $\mathcal{E}^{(r)}$ and is equivariant under the left action of the subgroup $\mathfrak{S}_r^{\mathrm{od}}\times\mathfrak{S}_r^{\mathrm{ev}}\subset\mathfrak{S}_{2r}$ on the group $\mathfrak{S}_{2r}$. Therefore the map $\mathfrak{S}_{2r}\to(\mathbb{F}_2)^{2r}$ factorizes to a surjective map
$$
\begin{equation}
\mathfrak{S}_r^{\mathrm{od}}\times\mathfrak{S}_r^{\mathrm{ev}} \setminus\mathfrak{S}_{2r}\to\mathcal{E}^{(r)}.
\end{equation}
\tag{4.6}
$$
The sets $\mathfrak{S}_r^{\mathrm{od}}\times\mathfrak{S}_r^{\mathrm{ev}}\setminus\mathfrak{S}_{2r}$ and $\mathcal{E}^{(r)}$ consist of the same number of elements. Hence the map (4.6) is a bijection. It remains to use the commutative diagram (4.5). Theorem 0.4 is proved. Finally, we note that Corollary 0.5 follows from Theorem 0.4 and the equality
$$
\begin{equation*}
\begin{aligned} \, &\begin{pmatrix} 1 & 2 & \dots & 2r-1 & 2r \\ i_1 & i_2 & \dots & i_{2r-1} & i_{2r} \end{pmatrix} \cdot \begin{pmatrix} 1 & 2 & \dots & 2r-1 & 2r \\ 2 & 3 & \dots & 2r & 1 \end{pmatrix} \\ &\qquad= \begin{pmatrix} 1 & 2 & \dots &2r-1 & 2r \\ i_2 & i_3 & \dots & i_{2r} & i_1 \end{pmatrix}. \end{aligned}
\end{equation*}
\notag
$$
4.3. The index codingThe description of deformation classes of real unmarked biquadrics (that is, intersections of two real quadrics) in [18], [22] uses the index coding. We now give an analogous description of the connected components of the variety of marked biquadrics. This description will be used in the proof of Theorem 0.6. We introduce the following equivalence relation on the space $\mathrm{S}^2(\mathbb{R}V^{\vee})$ of real quadratic forms: where $\lambda$ is a positive real number. The corresponding quotient space is a sphere. We denote it by $\mathbb{S}$. The equivalence class of a quadratic form $q$ is denoted by $[q]$. Then to every real biquadric $B\in\mathcal{BQ}^\circ$ there corresponds the unique great circle $S^1(B)$ through the points $[\mathfrak{q}], [-\mathfrak{q}]\in\mathbb{S}$. This circle is obtained from the pencil of quadratic forms determined by the biquadric. If $q$ and $q'$ are equivalent, then their positive indices of inertia coincide: $I_+(q)=I_+(q')$. Hence there is a well-defined index function $I\colon \mathbb{S}\to\mathbb{Z}$, $I([q])=I_+(q)$. We also have an index function $I_{B}\colon S^1(B)\to\mathbb{Z}$ equal to the restriction of the function $I\colon \mathbb{S}\to\mathbb{Z}$ to the circle $S^1(B)$.The index function $I_{B}$ satisfies the following conditions: 1) $I_{B}([\mathfrak{q}])=I_{B}([-\mathfrak{q}])=m$; 2) $I_{B}$ is lower semicontinuous; 3) $I_{B}$ takes integer values belonging to the interval $[0,n]$; 4) the value of $I_{B}$ jumps by $\pm1$ as we pass through a discontinuity point; 5) if $I_{B}$ is continuous at $[q]$, then $I_{B}([-q])=n-I_{B}([q])$; 6) if $I_{B}$ is discontinuous at $[q]$, then $I_{B}([-q])=n-I_{B}([q])-1$. Fixing an orientation of the circle $S^1(B)$, we have a well-defined semicircle arising as we move along $S^1(B)$ from $[-\mathfrak{q}]$ to $[\mathfrak{q}]$ in accordance with the orientation of $S^1(B)$. Note that the function $I_{B}$ is determined (up to a continuous transformation of this semicircle leaving the points $[\mathfrak{q}],[-\mathfrak{q}]\in\mathbb{S}$ fixed) by the sequence $\{m,i_1,\dots,i_{\nu-1},m\}$ of its values at continuity points arising as we move from $[-\mathfrak{q}]$ to $[\mathfrak{q}]$. If we reverse the orientation of $S^1(B)$, the sequence of values is replaced by the sequence of values We shall regard the sequences of values
$$
\begin{equation*}
\{m,i_1,\dots,i_{\nu-1},m\},\qquad \{m,n-i_{\nu-1},\dots,n-i_{1},m\}
\end{equation*}
\notag
$$
as equivalent. Then the equivalence class of a sequence $\{m,i_1,\dots,i_{\nu-1},m\}$ depends only on the deformation class of the biquadric $B$. Therefore we denote the equivalence class of $\{m,i_1,\dots,i_{\nu-1},m\}$ by and call it the index code of the biquadric $B$. Note that if $B\in\mathcal{BQ}^{(r)}$, then $\nu=2r$. Hence the index code of such a biquadric is of the form where $i_0=i_{2r}=m$. It is easy to see that the following proposition holds.Proposition 4.3. The index code of a biquadric $B$ uniquely determines the deformation class of $B$. Namely, let be the index code of a biquadric $B\in\mathcal{BQ}^{(r)}$. Then the deformation class of $B$ is where $e_k=0$ if $i_k<i_{k-1}$, and $e_k=1$ for all other $k=1,\dots,2r$. Remark. Using the properties of the index function, we see that the $\mathbb{F}_2$-code (that is, Rohn’s code reduced $\operatorname{mod}2$) of a biquadric determines the index function of this biquadric. This follows from Proposition 4.3. Then Corollary 0.5 yields a procedure for finding the index functions of cosingular biquadrics. Namely, we first construct the $\mathbb{F}_2$-code of a given biquadric, then list the $\mathbb{F}_2$-codes of all cosingular biquadrics and, finally, recover the index functions from these $\mathbb{F}_2$-codes. § 5. Rough projective classificationFirst of all, we note that the automorphism group $\operatorname{Aut}\mathfrak{Q}$ of the quadric $\mathfrak{Q}$ is equal to $\operatorname{PO}(V,\mathfrak{q})=\mathrm{O}(V,\mathfrak{q})/\{\pm1\}$, the projectivization of the group $\mathrm{O}(\mathfrak{q},V)$. In particular, every automorphism of $\mathfrak{Q}$ extends uniquely to an automorphism of the projective space $\mathbb{P}(V)$. 5.1. Rough projective classification of biquadricsWe write $\mathcal{A}ut\,\mathfrak{Q}$ for the subgroup of $\operatorname{Aut}\mathfrak{Q}$ consisting of all real automorphisms. Note that there is a left action of $\mathcal{A}ut\,\mathfrak{Q}$ on the set $\langle\mathcal{BQ}^{(r)}\rangle$ of connected components. The set of orbits
$$
\begin{equation*}
\mathcal{A}ut\,\mathfrak{Q}\setminus\langle\mathcal{BQ}^{(r)}\rangle
\end{equation*}
\notag
$$
is denoted by $\langle\!\langle\mathcal{BQ}^{(r)}\rangle\!\rangle$. These orbits are called rough projective classes of real biquadrics. It will be proved that every rough projective class consists of one or two deformation classes. We assume that $r>0$ since $\mathcal{BQ}^{(0)}$ has a single component. Let $\mathcal{PO}(\mathfrak{q},V)$ be the projectivization of $\mathcal{O}(\mathfrak{q},V)$, that is,
$$
\begin{equation*}
\mathcal{PO}(\mathfrak{q},V)=\mathcal{O}(\mathfrak{q},V)/\{\pm1\}.
\end{equation*}
\notag
$$
Lemma 5.1. The group $\mathcal{PO}(\mathfrak{q},V)$ acts trivially on $\langle \mathcal{BQ}^{(r)}\rangle$. Proof. The group $\mathcal{O}(\mathfrak{q},V)$ consists of four connected components (see § 2.2), which were denoted by $\mathcal{O}(\mathfrak{q},V)_{\pm\pm}$. Elements of $\mathcal{O}(\mathfrak{q},V)_{++}$ do not interchange the components of $\mathcal{BQ}^{(r)}$ since $\mathcal{O}(\mathfrak{q},V)_{++}$ contains the identity transformation. If $B\in\mathcal{BQ}^{(r)}$, then there is a system of coordinates $(v_1,\dots,v_n)\in\mathcal{SK}^{(r)}$ such that the biquadric $B$ is given by the system of equations Since $r>0$, the coordinate $v_1$ is real and $v_2$ is imaginary. Multiplying one or both of these coordinates by $-1$, we obtain transformations belonging to the components respectively. Since these transformations preserve $B$, they preserve the component of $\mathcal{BQ}^{(r)}$ containing $B$. $\Box$ We now give an example of an element of $\mathcal{A}ut\,\mathfrak{Q}$ not belonging to $\mathcal{PO}(\mathfrak{q},V)$. Fix a coordinate system $\operatorname{sk}=(v_1,\dots,v_n)\in\mathcal{SK}^{(r)}$ and define transformations
$$
\begin{equation*}
h=h_{\operatorname{sk}}\colon V\to V,\qquad h'=h'_{\operatorname{sk}}\colon V\to V
\end{equation*}
\notag
$$
by the equalities
$$
\begin{equation*}
\begin{aligned} \, h(v_1,\dots,v_n) &= (iv_2,iv_1,\dots,iv_{2r},iv_{2r-1},iv_{2r+1},-iv_{2r+2},\dots,iv_{n-1},-iv_{n}), \\ h'(v_1,\dots,v_n) &= (v_2,v_1,\dots,v_{2r},v_{2r-1},v_{2r+1},-v_{2r+2},\dots,v_{n-1},-v_{n}), \end{aligned}
\end{equation*}
\notag
$$
where $i=\sqrt{-1}$. The transformation $h$ is real and $h'$ belongs to $\mathrm{O}(V,\mathfrak{q})$. Denoting the projectivizations of $h$, $h'$ by $[h]$, $[h']$, we obtain an equality $[h]=[h']$. Hence $[h]$ belongs to $\mathcal{A}ut\,\mathfrak{Q}$. On the other hand, since $h$ multiplies the quadratic form $\mathfrak{q}(\mathbf{v})$ by $-1$, the projective automorphism $[h]$ does not belong to $\mathcal{PO}(\mathfrak{q},V)$. Proposition 5.2. The group $\mathcal{PO}(\mathfrak{q},V)$ is a normal subgroup of $\mathcal{A}ut\,\mathfrak{Q}$. The quotient group $\mathcal{A}ut\,\mathfrak{Q}/\mathcal{PO}(\mathfrak{q},V)$ is isomorphic to $\mathbb{Z}/2\mathbb{Z}$. Proof. Let $C$ be the group of order two generated by the involution $c\colon V\to V$. This group acts on the groups $\mathrm{O}(V,\mathfrak{q})$ and $\operatorname{PO}(V,\mathfrak{q})$, and the exact sequence
$$
\begin{equation*}
1\to\{\pm1\}\to\mathrm{O}(V,\mathfrak{q})\to\operatorname{PO}(V,\mathfrak{q})\to1
\end{equation*}
\notag
$$
yields an exact sequence of groups that is, an exact sequence Hence $\mathcal{PO}(\mathfrak{q},V)$ is a normal subgroup of $\mathcal{A}ut\,\mathfrak{Q}$ and, moreover, the quotient group $\mathcal{A}ut\,\mathfrak{Q}/\mathcal{PO}(\mathfrak{q},V)$ is either isomorphic to $\mathbb{Z}/2\mathbb{Z}$ or equal to zero. The example above shows that the first option holds. $\Box$ In what follows we denote the group $\mathcal{PO}(\mathfrak{q},V)$ by $\mathcal{A}ut_0\,\mathfrak{Q}$. Since the transformation
$$
\begin{equation*}
[h]\in\mathcal{A}ut\,\mathfrak{Q}\setminus\mathcal{A}ut_0\,\mathfrak{Q}
\end{equation*}
\notag
$$
is an involution, we see from Proposition 5.2 and Lemma 5.1 that a rough projective class of biquadrics contains at most two deformation classes. The following lemma can easily be proved. Lemma 5.3. Suppose that $\operatorname{sk}\in\mathcal{SK}^{(r)}$, $h=h_{\operatorname{sk}}$ is the corresponding transformation and $B\in\mathcal{BQ}^\circ(\operatorname{sk})$. If the index code of $B$ is then the biquadric $[h](B)$ has index code We define an equivalence relation on the set $\mathcal{E}^{(r)}/\Sigma(2r)$ of $\mathbb{F}_2$-codes by declaring that every $\mathbb{F}_2$-code $|e_1,\dots,e_{2r}|$ is equivalent to the $\mathbb{F}_2$-code obtained from $|e_1,\dots,e_{2r}|$ by replacing zeros by ones and, vice versa, ones by zeros. The set of equivalence classes is denoted by $\langle\mathcal{E}^{(r)}/\Sigma(2r)\rangle$, and the equivalence class of an $\mathbb{F}_2$-code $|e_1,\dots,e_{2r}|$ is denoted by $\|e_1,\dots,e_{2r}\|$. Using Proposition 4.3 and Lemma 5.3, we see that Theorem 5.4 holds. Theorem 5.4. The bijection $\langle\mathcal{BQ}^{(r)}\rangle=\mathcal{E}^{(r)}/\Sigma(2r)$ induces a bijection 5.2. Rough projective classification of Kummer varietiesThe group $\operatorname{PSO}(\mathfrak{q},V)$ acts on the orthogonal Grassmannians $\operatorname{G}_\pm$. Hence there are group homomorphisms
$$
\begin{equation*}
\operatorname{PSO}(\mathfrak{q},V)\to\operatorname{Aut}\operatorname{G}_\pm.
\end{equation*}
\notag
$$
The following theorem holds (see [25], [11]). Theorem 5.5. If $n\geqslant6$ and $n\neq8$, then the homomorphisms $\operatorname{PSO}(\mathfrak{q},V)\to\operatorname{Aut}\operatorname{G}_\pm$ are isomorphisms. When $n=8$, the Grassmannians $\operatorname{G}_\pm$ are isomorphic to a four-dimensional quadric (see [15]) and, therefore, the automorphism groups $\operatorname{Aut}\operatorname{G}_\pm$ consist of two connected components. Hence, when $n=8$, the homomorphisms
$$
\begin{equation*}
\operatorname{PSO}(\mathfrak{q},V)\to\operatorname{Aut}\operatorname{G}_\pm
\end{equation*}
\notag
$$
are not isomorphisms but each of them maps $\operatorname{PSO}(\mathfrak{q},V)$ isomorphically onto the component containing the identity transformation. Let $\mathcal{A}ut\operatorname{G}_\pm$ be the subgroup of $\operatorname{Aut}\operatorname{G}_\pm$ consisting of all real automorphisms, and let $\mathcal{A}ut(\mathfrak{Q},+)$ be the group $\mathcal{A}ut\,\mathfrak{Q}\,\cap\,\operatorname{PSO}(\mathfrak{q},V)$. Note that the automorphism $[h]$ (constructed above) belongs to $\mathcal{A}ut(\mathfrak{Q},+)$. Thus we obtain homomorphisms of real groups
$$
\begin{equation*}
\mathcal{A}ut(\mathfrak{Q},+)\to\mathcal{A}ut\operatorname{G}_\pm,
\end{equation*}
\notag
$$
which are isomorphisms for $n\geqslant6$, $n\neq8$. In what follows we assume that $n\neq8$, fix one of the varieties $\operatorname{G}_\pm$ and denote it by $\operatorname{G}$ and the corresponding automorphism group by $\mathcal{A}ut\operatorname{G}$. There is a left action of $\mathcal{A}ut\operatorname{G}$ on $\langle\mathcal{KV}^{(r)}\rangle$. The set of orbits
$$
\begin{equation*}
\mathcal{A}ut\operatorname{G}\setminus\langle\mathcal{KV}^{(r)}\rangle
\end{equation*}
\notag
$$
is denoted by $\langle\!\langle\mathcal{KV}^{(r)}\rangle\!\rangle$ and is called the set of rough projective classes of real Kummer varieties. Besides the bijection in Theorem 5.4, for $n\neq8$ we have a canonical bijection
$$
\begin{equation}
\langle\!\langle\mathcal{KV}^{(r)}\rangle\!\rangle \xrightarrow{\approx}\langle\mathcal{E}^{(r)}/\Delta(2r)\rangle,\qquad r>0,
\end{equation}
\tag{5.1}
$$
where the set $\langle\mathcal{E}^{(r)}/\Delta(2r)\rangle$ is defined similarly to $\langle\mathcal{E}^{(r)}/\Sigma(2r)\rangle$. Namely, we define an equivalence relation on the set $\mathcal{E}^{(r)}/\Delta(2r)$ of $\mathbb{F}_2$-codes by declaring that every $\mathbb{F}_2$-code $[e_1,\dots,e_{2r}]$ is equivalent to the $\mathbb{F}_2$-code obtained from $[e_1,\dots,e_{2r}]$ by replacing zeros by ones and, vice versa, ones by zeros. The set of equivalence classes is denoted by $\langle\mathcal{E}^{(r)}/\Delta(2r)\rangle$ and the equivalence class of an $\mathbb{F}_2$-code $[e_1,\dots,e_{2r}]$ is denoted by $[[e_1,\dots,e_{2r}]]$. The bijection (5.1) is induced by the bijection
$$
\begin{equation*}
\langle\mathcal{KV}^{(r)}\rangle \xrightarrow{\approx}\mathcal{E}^{(r)}/\Delta(2r),\qquad r>0.
\end{equation*}
\notag
$$
This fact follows from Corollary 0.5 and Theorem 5.4. As a result, we see that Corollary 5.6 holds for $r>0$, $n\neq8$. Remark. It is possible that the sets $\langle\!\langle\mathcal{KV}^{(r)}\rangle\!\rangle$ and $\langle\mathcal{E}^{(r)}/\Delta(2r)\rangle$ do not coincide for $n=8$. This is a separate problem in the study of the case when $n=8$. We do not consider it here. 5.3. Classification for $n=6$The following proposition for three-dimensional biquadrics can be obtained from general results by direct verification. For brevity, we omit the commas between the zeros and ones in $\mathbb{F}_2$-codes. Proposition 5.7. When $n=6$, the rough projective classification of Kummer varieties coincides with their deformation classification. Thus, $\langle\!\langle\mathcal{KV}^\circ\rangle\!\rangle= \langle\mathcal{KV}^\circ\rangle$. Proof. Projectively equivalent components of $\mathcal{BQ}^\circ$ form the following four and only four pairs: Therefore we have Remark. The equality in Proposition 5.7 does not hold for $n>6$. To prove this, we consider the following pair of projectively equivalent components of $\mathcal{BQ}^{(4)}$: The codes $|01100011|$, $|10011100|$ determine the same point of $\langle\mathcal{E}^{(r)}/\Delta(2r)\rangle$ but distinct points of $\mathcal{E}^{(r)}/\Delta(2r)$. 5.4. Classifications of real quadratic complexes and the associated Kummer quarticsThree-dimensional marked biquadrics whose marked quadric is the Grassmannian (embedded in $\mathbb{P}^5$ by the Plücker map) of lines in $\mathbb{P}^3$ are called quadratic line complexes. Note that this marked quadric is given by the equation $\mathfrak{q}(\mathbf{v})=0$, where $\mathfrak{q}(\mathbf{v})$ is the Plücker–Klein quadratic form. The restriction of this form to the real part is of signature $(3,3)$. Therefore Proposition 5.7 is actually an assertion about Kummer quartics. Rohn [5] uses permutations in order to partition the set of real quadratic complexes into classes. This partition and the Plücker–Klein map are used in [5] to partition the set of real Kummer quartics into classes. The notions of deformation classification and rough projective classification did not exist at the time of creation of [5]. But Rohn has actually constructed a rough projective classification of real quadratic complexes. The induced partition of the set of real Kummer quartics yields a rough projective classification of these quartics, which coincides with their deformation classification (see the fifth part of Proposition 5.7). This means that all real Kummer quartics are chiral, that is, the mirror image of a Kummer quartic belongs to its deformation class. The author [7], [8] obtained a classification of real quadratic complexes by using the index coding. The author was unaware of Rohn’s paper [5] while writing those papers. They actually contain a description of the set $\langle\!\langle\mathcal{\mathcal{BQ}^\circ}\rangle\!\rangle$ for $n=6$, that is, a rough projective classification of real non-singular quadratic complexes, which was mistakenly referred to as a rigid isotopic (deformation) classification. Since replacing a quadratic complex by a projectively equivalent complex does not change the deformation class of the associated Kummer quartic (see Proposition 5.7), the mistake in [7], [8] does not result in any mistakes in the deformation classification of real Kummer quartics in [8]. However, the reader should have in mind that the phrase ‘rigid isotopic (deformation) classes of quadratic complexes’ is to be replaced by the phrase ‘rough projective classes of quadratic complexes’ in all the theorems in [7], [8] and the subsequent papers by the author (see, for example, Theorem 3.1 in [10], which speaks of connected components but actually makes assertions about rough projective classes). More details about this mistake can be found in the remark at the end of this section. The classical real Plücker–Klein map was used in the study of real Kummer quartics. In particular, Plücker, Klein, and Rohn used it to construct models of the real parts of Kummer quartics (see [26]). Note that the topological structure of these parts can easily be described by factorizing the real parts of real Abelian surfaces (see Theorem 10.5 and more details in § 7). Rohn’s corresponding results in [5], which were obtained by using the Plücker–Klein map, were stated inaccurately, but the models constructed show that the intuitive understanding was correct. Corollary 5.8. Real biquadrics $B,B'\in\mathcal{BQ}^\circ$ belong to the same connected component of $\mathcal{BQ}^\circ$ if and only if there is a homeomorphism of the circle $S^1(B)$ onto the circle $S^1(B')$ which is equivariant under the antipodal involution. sends the index function $I_{B}$ to the index function $I_{B'}$ and leaves the points fixed. The statements of analogues of Corollary 5.8 in [7], [8] erroneously admit homeomorphisms that interchange the points $[\mathfrak{q}]$, $[-\mathfrak{q}]$ (see [7], Proposition 1.3, and [8], Proposition 2.3). Since both equations
$$
\begin{equation*}
\mathfrak{q}(\mathbf{v})=0,\qquad -\mathfrak{q}(\mathbf{v})=0
\end{equation*}
\notag
$$
determine the quadric $\mathfrak{Q}$, the author erroneously admitted homeomorphisms that interchange the points $[\mathfrak{q}]$, $[-\mathfrak{q}]$. The group of real linear transformations of $V$ that send the quadric $\mathfrak{Q}$ to itself contains transformations that send the form $\mathfrak{q}$ to $-\mathfrak{q}$, but they do not belong to the connected component of the identity transformation. Hence such transformations may send one connected component of $\mathcal{BQ}^\circ$ to another. And this indeed occurs.
§ 6. The family of Kummer varieties of a biquadricWe state some definitions and results from [12], [27] that are needed in the proof of Theorem 0.6. 6.1. The family of orthogonal GrassmanniansIf $Q\subset\mathbb{P}(V)$ is a singular quadric with a unique singular point, then the Fano variety $\Phi(Q)$ of all $g$-planes in $Q$ is connected. Suppose that $B\in\operatorname{BQ}^\circ$ and let $L=L(B)$ be the corresponding pencil of quadrics. We write $\operatorname{Gr}$ for the ordinary Grassmannian of all $g$-planes in $\mathbb{P}(V)$. Let $\Phi(L)$ be the incidence subvariety of $L\times\operatorname{Gr}$. It consists of the pairs $(p,\Pi)$, where $\Pi$ is a $g$-plane on the quadric $Q_p$ in the pencil $L$. We write $\operatorname{pr}\colon \Phi(L)\to L$ for the projection onto the first factor. When $p\in L$ is one of the points $p_1,\dots,p_n$ corresponding to the singular quadrics, the fibre $\operatorname{pr}^{-1}(p)$ consists of a single connected component. Otherwise it consists of two components. Performing the Stein factorization of points in the components of the fibres $\operatorname{pr}^{-1}(p)$, we obtain a canonical double covering of the line $L$ branched at the points $p_1,\dots,p_n\in L$. We denote it by $\pi\colon W\to L$. Since it also depends on the biquadric $B$, the notation $\pi\colon W(B)\to L(B)$ is more complete. Note that the Stein factorization induces a projection $\operatorname{pr}\colon \Phi(L)\to W$ sending every point $(p,\Pi)$ to its equivalence class. The fibres of this projection are orthogonal Grassmannians, which are isomorphic to each other. Thus we obtain a locally trivial bundle into orthogonal Grassmannians. We denote it by $\operatorname{pr}\colon \operatorname{G}(W)\to W$. 6.2. The maximal Fano variety of a biquadricWe recall that a compact complex manifold $X$ acted on freely and transitively by a complex torus $A$ is called an $A$-torsor. Taking a fixed point $x_0$ of an $A$-torsor $X$ for zero, we obtain a canonical isomorphism $A=X$ by the rule where $t_a$ is the shift (translation) given by the action of $A$ on $X$.If $B\in\operatorname{BQ}^\circ$, then $B$ contains no $g$-planes but there are $(g-1)$-planes. We write $F=F(B)$ for the Fano variety of all $(g-1)$-planes on $B$. It is a smooth projective variety of dimension $g$. We call it the maximal Fano variety of the biquadric $B$. It is known that $F(B)$ is isomorphic to the Jacobian $J=J(B)$ of the curve $W=W(B)$ (see [12]). The isomorphism $F=J$ constructed in [12] is not uniquely determined but one can see that it is determined up to a translation. Therefore this isomorphism endows $F$ with the structure of a $J$-torsor. In what follows we denote points of $F$ by $x$, $y$, $o$, $\dots$ . Taking a fixed point $o\in F$ for the zero element, we obtain a group law on $F$, which will be denoted by We also obtain the inversion involution (multiplication by $-1$) denoted by On the other hand, for every $w\in W$ we have the following involution $i_w\colon F\to F$. If $x\in F$, then there is a unique $g$-plane $s\in\operatorname{G}_w$ containing $x$ (see [12]). The intersection $s\cap B$ consists of $x\in F$ and a complementary $(g-1)$-plane $x'\in F$ (possibly coinciding with $x$). By definition, we put $i_w(x)=x'$.We now complement the geometric definition of the involution $i_w\colon F\to F$ by an algebraic formula. Let $F^{(w)}$ be the set of fixed points of $i_w\colon F\to F$, that is, $F^{(w)}=F^{i_w}$. This set is non-empty. We fix a point $o\in F^{(w)}$ and take it for the zero element of the torsor $F$. Then it turns out that $i_w=(-1_o)$ (see [12]). We shall need the following property of the involutions $i_w$. Let $t_{w,w'}\colon F\to F$ be the translation transformation given by the divisor class $(w'-w)$, where $w,w'\in W$. We write $K_w$ for the geometric quotient $F/i_w$. Since $i_w=(-1_o)$, where $o\in F^{(w)}$, it follows that $K_w$ is a Kummer variety. Let $\pi_w\colon F\to\operatorname{G}_w$ be the map sending every $(g-1)$-plane $P\in F$ to the unique $g$-plane $\Pi\in\operatorname{G}_w$ containing $P$. The map $\pi_w\colon F\to\operatorname{G}_w$ factorizes to a map $p_w\colon K_w\to\operatorname{G}_w$, which is an embedding. We shall identify $K_w$ with its image in $\operatorname{G}_w$. The family of all Kummer varieties $K_w\subset\operatorname{G}_w$ is a subvariety $\operatorname{K}(W)\subset\operatorname{G}(W)$. It follows from Proposition 6.1 that the corresponding projection $\operatorname{pr}\colon \operatorname{K}(W)\to W$ induces a locally trivial bundle. 6.3. A special embedding of $W$ in $F$Let $*\in W$ be a fixed branch point of the curve $W$. Then we take a point $o$ in the set $F^{(*)}$ for the zero element. Let $\varphi_o\colon W\to F$ be the map sending every point $w\in W$ to the point $i_w(o)\in F$. The image of $W$ under this map is denoted by $W_o$. The map $\varphi_o\colon W\to W_o$ is an isomorphism. It turns out that the curve $W_o$ is invariant under the involution $(-1_o)\colon F\to F$, and the restriction of this involution to $W_o$ coincides with the hyperelliptic involution $\sigma\colon W_o\to W_o$ (see [27]). For a fixed branch point $*\in W$ we consider the Abel–Jacobi map $AJ_*\colon W\to J$ sending every point $w\in W$ to the divisor class $(w-*)$. This map is an embedding. We identify the curve $W$ with its image under the Abel–Jacobi map. Then the following theorem holds (see [27]). Theorem 6.2. The isomorphism $\varphi_o\colon W\xrightarrow{\approx}W_o$ extends uniquely to a group isomorphism $\varphi_o\colon (J,+)\to (F,+_o)$ of Abelian varieties. § 7. Real tori and Kummer varietiesIn this section we state some results from [28]–[33], [9] which are needed in the study of the topology of real Kummer varieties arising from real biquadrics. Let $\mathcal{L}$ be a $g$-dimensional real vector space, $g\geqslant2$, and let $\Lambda\subset\mathcal{L}$ be a lattice of rank $g$. Then the quotient space $\mathcal{L}/\Lambda$ is denoted by $\mathcal{T}=\mathcal{T}_{\mathcal{L},\Lambda}$ and is called a topological torus. Note that $\mathcal{T}$ is a compact commutative Lie group. Let $(-1)\colon \mathcal{T}\to\mathcal{T}$ be the inversion involution. Then the quotient space $\mathcal{T}/(-1)$ is denoted by $\mathcal{K}=\mathcal{K}(\mathcal{T})$ and is called a topological Kummer space. Since $\mathcal{K}$ contains points whose neighbourhoods are not diffeomorphic to a ball, $\mathcal{K}$ is not a differentiable manifold but merely a differentiable orbifold. These points arise from points of order 2 in the group $\mathcal{T}$ and are called singular points of $\mathcal{K}$. The neighbourhood of a singular point in $\mathcal{K}$ is diffeomorphic to a cone with base $\mathbb{RP}^{g-1}$. When $g=2$, the base of this cone is homeomorphic to a circle and, therefore, $\mathcal{K}$ is a topological manifold (a two-dimensional topological sphere). When $g>2$, $\mathcal{K}$ is not a topological manifold. Note that any two topological Kummer spaces of equal dimensions are diffeomorphic. Therefore every topological Kummer space of dimension $g$ is denoted by $\mathcal{K}^g$. Note that the number of singular points of $\mathcal{K}^g$ is equal to $2^g$. We shall use the following standard model of $\mathcal{K}^g$. Put $\mathcal{L}=\mathbb{R}^g$ and $\Lambda=\mathbb{Z}^g$. Then and the set of singular points of $\mathcal{K}^g$ is the $\mathbb{F}_2$-vector space
$$
\begin{equation*}
\biggl(\frac{1}{2}\mathbb{Z}/\mathbb{Z}\biggr)^g=(\mathbb{F}_2)^g.
\end{equation*}
\notag
$$
Let $L$ be a $g$-dimensional complex vector space with an antilinear involution $\theta\colon L\to L$, where $g\geqslant2$, and let $\Lambda\subset L$ be a lattice of rank $2g$. The lattice $\Lambda$ is said to be real if it is invariant under the involution $\theta\colon L\to L$. Given such a lattice $\Lambda$, we write $\Lambda^\pm$ for the sublattices
$$
\begin{equation*}
\operatorname{Ker}[1\mp\theta\colon \Lambda\to \Lambda].
\end{equation*}
\notag
$$
Then the quotient group is a $2$-periodic group, that is, an $\mathbb{F}_2$-space. We denote the dimension of this $\mathbb{F}_2$-space by $d=d(\Lambda)$. If $\Lambda\subset L$ is a real lattice, then the involution $\theta$ descends to the complex torus $L/\Lambda$. The pair $(L/\Lambda,\theta)$ is called a real torus and the antiholomorphic involution is called the real structure of this real torus.Let $\Lambda\subset L$ be a real lattice. Then the real torus $(L/\Lambda,\theta)$ is denoted by $A=A_{L,\Lambda}$ and we write $A^\pm=A_{L,\Lambda}^\pm$ for the Lie subgroups The connected components of zero in $A^\pm$ are denoted by $A_0^\pm$. The varieties $A_0^\pm$ are topological tori of dimension $g$. It turns out that there are Lie group isomorphisms (see, for example, [28]–[30])
$$
\begin{equation}
A^\pm\cong A^\pm_0\oplus(\mathbb{Z}/2\,\mathbb{Z})^{d(\Lambda)}.
\end{equation}
\tag{7.1}
$$
Note that these isomorphisms are not canonical. The isomorphisms (7.1) show that every variety $A^\pm$ consists of $2^d$ topological tori of dimension $g$. We claim that the connected components of $A^+$ are enumerated by the elements of the cohomology group $H^1(\Theta,\Lambda)$, where $\Theta=\{1,\theta\}$ is a group of order two. Indeed, the short exact sequence of $\Theta$-modules yields a long exact sequence of cohomology groups of $\Theta$. This sequence in turn yields a canonical isomorphism (details can be found, for example, in [31])The groups $\{1,\theta\}$ and $\{1,-\theta\}$ will also be denoted by $\Theta^+$ and $\Theta^-$ respectively. Then we obtain a canonical isomorphism Given any $\eta\in H^1(\Theta^\pm,\Lambda)$, we denote the corresponding topological tori by $\mathcal{A}^\pm_\eta$. The groups $H^1(\Theta^+,\Lambda)$ and $H^1(\Theta^-,\Lambda)$ are isomorphic to each other since they are both isomorphic to $(\mathbb{Z}/2\,\mathbb{Z})^d$. Note that the number $d=d(\Lambda)$ is uniquely determined by the real torus $(A,\theta)$ and, therefore, we also denote it by $d(A)$. The number $d(A)$ satisfies the inequality $d(A)\leqslant g$ and may take any integer value from $0$ to $g$.Note that if $M$ is an arbitrary $\Theta$-module, then
$$
\begin{equation}
H^q(\Theta^\pm,M)= \begin{cases} \operatorname{Ker}[1\mp\theta\colon M\to M] &\text{for }q=0, \\ \operatorname{Ker}[1\pm\theta\colon M\to M]/\operatorname{Im}[1\mp\theta\colon M\to ] &\text{for }q=2k+1>0, \\ \operatorname{Ker}[1\mp\theta\colon M\to M]/\operatorname{Im}[1\pm\theta\colon M\to M] &\text{for }q=2k>0. \end{cases}
\end{equation}
\tag{7.2}
$$
The intersection $A^+\cap A^-$ is equal to the set of points of order two in $A$. This set is an $\mathbb{F}_2$-vector space of dimension $g\,{+}\,d$. Hence the number of points in $A^+\cap A^-$ is equal to $2^{g+d}$. For any elements
$$
\begin{equation*}
\mu\in H^1(\Theta^+,\Lambda),\qquad \nu\in H^1(\Theta^-,\Lambda)
\end{equation*}
\notag
$$
the topological tori $\mathcal{A}^+_\mu$, $\mathcal{A}^-_\nu$ have $2^{g-d}$ common points. If $a\in\mathcal{A}^+_\mu\cap\mathcal{A}^-_\nu$, then
$$
\begin{equation*}
\mathcal{A}^+_\mu=A^+_0+a,\qquad \mathcal{A}^-_\nu=A^-_0+a.
\end{equation*}
\notag
$$
Let $(-1)\colon A\to A$ be the inversion involution. Then we write $K=K(A)$ for the geometric quotient $A/(-1)$ and denote the corresponding projection by $\pi\colon A\to K$. Since the involutions $\theta$ and $(-1)$ commute, the real structure $\theta\colon A\to A$ descends to a real structure $\theta\colon K\to K$. The real part $\mathbb{R}K$ of the variety $K$ is equal to $\pi(A^+)\cup\pi(A^-)$. Given any $\eta\in H^1(\Theta^\pm,\Lambda)$, we write $\mathcal{K}^\pm_\eta$ for the topological Kummer varieties arising from the topological tori $\mathcal{A}^\pm_\eta$. Then the real part of the Kummer variety $K(A)$ is equal to the union of the topological Kummer spaces $\mathcal{K}^\pm_\eta$. For any elements
$$
\begin{equation*}
\mu\in H^1(\Theta^+,\Lambda),\qquad \nu\in H^1(\Theta^-,\Lambda),
\end{equation*}
\notag
$$
the topological Kummer spaces $\mathcal{K}^+_\mu$ and $\mathcal{K}^-_\nu$ intersect each other at $2^{g-d}$ singular points. Hence the real part $\mathbb{R}K$ of the Kummer variety $K$ consists of $2^{d+1}$ topological Kummer spaces $\mathcal{K}^g$. To understand the structure of $\mathbb{R}K$ completely, we need to describe common points of $\mathcal{K}^+_\mu$ and $\mathcal{K}^-_\nu$. Here is the required description. If $d(\Lambda)=d$, then the lattice $\Lambda$ has a basis $\lambda_1,\dots, \lambda_{2g}$ such that the vectors $\lambda_2, \lambda_4, \dots, \lambda_{2d}$ are real, the vectors $\lambda_{1}, \lambda_{3}, \dots, \lambda_{2d-1}$ are imaginary and every pair $\{\lambda_{2d+1},\lambda_{2d+2}\}, \dots, \{\lambda_{2g-1},\lambda_{2g}\}$ consists of complex-conjugate vectors (see, for example, [32]). Hence the real part $\Lambda^+$ of the lattice $\Lambda$ is generated by the vectors
$$
\begin{equation*}
\lambda_2,\lambda_4,\dots,\lambda_{2d},\qquad \lambda_{2d+1}+\lambda_{2d+2},\dots,\lambda_{2g-1}+\lambda_{2g}
\end{equation*}
\notag
$$
and the imaginary part $\Lambda^-$ of $\Lambda$ is generated by the vectors
$$
\begin{equation*}
\lambda_{1},\lambda_{3},\dots,\lambda_{2d-1},\qquad \lambda_{2d+1}-\lambda_{2d+2},\dots,\lambda_{2g-1}-\lambda_{2g}.
\end{equation*}
\notag
$$
Note that the group ${}_2A$ of points of order 2 is isomorphic to the quotient group $\frac{1}{2}\Lambda/\Lambda$. Hence the subgroup of real points of order 2 in ${}_2A$ is equal to the fixed part of the involution
$$
\begin{equation*}
\theta\colon \frac{1}{2}\Lambda/\Lambda\to\frac{1}{2}\Lambda/\Lambda.
\end{equation*}
\notag
$$
We denote this subgroup by ${}_2A^\theta$ and consider the following elements of it:
$$
\begin{equation*}
\begin{gathered} \, \begin{alignedat}{7} e_1^+ &=\frac{1}{2}\lambda_2 \ \operatorname{mod}\Lambda, &\quad e_2^+&=\frac{1}{2}\lambda_4 \ \operatorname{mod}\Lambda, &\quad &\dots, &\quad e_{d}^+&=\frac{1}{2}\lambda_{2d}\ \operatorname{mod}\Lambda, \\ e_1^- &=\frac{1}{2}\lambda_{1}\ \operatorname{mod}\Lambda, &\quad e_2^- &=\frac{1}{2}\lambda_{3} \ \operatorname{mod}\Lambda, &\quad &\dots, &\quad e_{d}^-&=\frac{1}{2}\lambda_{2d-1}\ \operatorname{mod}\Lambda, \end{alignedat} \\ \begin{alignedat}{5} h_1^+ &=\frac{1}{2}(\lambda_{2d+1}+\lambda_{2d+2})\ \operatorname{mod}\Lambda, &\quad &\dots, &\quad h_{g-d}^+&= \frac{1}{2}(\lambda_{2g-1}+\lambda_{2g})\ \operatorname{mod}\Lambda, \\ h_1^- &=\frac{1}{2}(\lambda_{2d+1}-\lambda_{2d+2})\ \operatorname{mod}\Lambda, &\quad &\dots, &\quad h_{g-d}^- &=\frac{1}{2}(\lambda_{2g-1}-\lambda_{2g})\ \operatorname{mod}\Lambda. \end{alignedat} \end{gathered}
\end{equation*}
\notag
$$
Then we have Therefore the elements $h_1^\pm,\dots,h_{g-d}^\pm$ are denoted by $h_1,\dots,h_{g-d}$. Let $E^\pm$ (resp. $H$) be the subgroups of ${}_2A^\theta$ generated by the elements $e_1^\pm,\dots,e_d^\pm$ (resp. $h_1,\dots,h_{g-d}$). When $d=g$, we put $H=\{0\}$ by definition. Thus, Writing $L^\pm$ for the real and imaginary parts of the vector space $L$, we have
$$
\begin{equation*}
A^\pm_0=L^\pm/\Lambda^\pm,\qquad {}_2A^\pm_0=E^\pm\oplus H.
\end{equation*}
\notag
$$
Moreover, the following equalities hold: If $\mu\in E^+$ (resp. $\nu\in E^-$), then we write $\mathcal{A}^-_{\mu}$ (resp. $\mathcal{A}^+_{\nu}$) for the topological torus $A^-_0+\mu$ (resp. $A^+_0+\nu$). Thus we obtain an equality
$$
\begin{equation*}
\mathcal{A}^+_{\nu}\cap\mathcal{A}^-_{\mu}=H+\nu+\mu,
\end{equation*}
\notag
$$
which yields the following formula:
$$
\begin{equation}
\mathcal{K}^+_{\nu}\cap\mathcal{K}^-_{\mu}=t_{\mu+\nu}(H),
\end{equation}
\tag{7.3}
$$
where $\mathcal{K}^+_{\nu}=\mathcal{A}^+_{\nu}/(-1)$, $\mathcal{K}^-_{\mu}=\mathcal{A}^-_{\mu}/(-1)$ and $t_{\mu+\nu}\colon K(A)\to K(A)$ is the translation by the point $\mu+\nu\in A$ of order 2. The formula (7.3) is the desired description of the intersection $\mathcal{K}^+_{\nu}\cap\mathcal{K}^-_{\mu}$. Every such intersection is a coset of the subgroup $H$ in the group ${}_2A^\theta$. Note that real lattices $\Lambda_1,\Lambda_2\subset L$ are deformation equivalent if and only if $d(\Lambda_1)=d(\Lambda_2)$ (see [9]). Therefore the spaces $\mathbb{R}K(L/\Lambda_1)$ and $\mathbb{R}K(L/\Lambda_2)$ are homeomorphic for any real lattices $\Lambda_1,\Lambda_2\subset L$ with $d(\Lambda_1)=d(\Lambda_2)$. We now present a standard model $\mathcal{K}^g_d$, $d=0,\dots,g$, of the topological space $\mathbb{R}K(L/\Lambda)$, where $d(\Lambda)=d$. The $\mathbb{F}_2$-space $(\mathbb{F}_2)^g$ of singular points of the standard model $\mathcal{K}^g$ of a topological Kummer space contains subspaces $E\,{\subset}\,(\mathbb{F}_2)^g$ and $H\subset(\mathbb{F}_2)^g$, where $E$ (resp. $H$) consists of the vectors all of whose coordinates with numbers greater than $d$ (resp. $\leqslant d$) are equal to zero. For every vector $e\in E$ we denote the coset $H+e$ by $H_e$. Consider the set of pairs $\{\mathcal{K}^+_\mu,\mathcal{K}^-_\mu\}$. It consists of $2^d$ pairs of standard models $\mathcal{K}^g$ enumerated by the vectors in $E$. For every such pair $\mathcal{K}^\pm_\mu$ there are pairs $H_{\mu,e}^\pm$ of cosets. We write $\widetilde{\mathcal{K}}^g_d$ for the disjoint union
$$
\begin{equation*}
\biggl(\bigsqcup_\mu\mathcal{K}^+_\mu\biggr)\sqcup\biggl(\bigsqcup_\nu\mathcal{K}^-_\nu\biggr).
\end{equation*}
\notag
$$
For every pair of vectors $\mu,\nu\in E$ we have canonical bijections $H_{\mu,\nu}^+=H$ and $H_{\nu,\mu}^-=H$. Hence we obtain a canonical bijection Note that the desired topological space $\mathcal{K}^g_d$ is obtained from the topological space $\widetilde{\mathcal{K}}^g_d$ by using the bijection described above in order to identify the sets of singular points $H_{\mu,\nu}^+\subset\mathcal{K}^+_\mu$ and $H_{\nu,\mu}^-\subset\mathcal{K}^-_\nu$ for every pair of vectors $\mu,\nu\in E$. This assertion follows from (7.3). Thus we obtain Proposition 7.1. Proposition 7.1. If $d(\Lambda)=d$, then the topological space $\mathbb{R}K(L/\Lambda)$ is homeomorphic to the topological space $\mathcal{K}^g_d$ just constructed. The real structure $\theta\colon A\to A$ can be twisted by a shift. Namely, given any $\eta\in A$, we write $\theta_\eta\colon A\to A$ for the anitholomorphic transformation $\theta_\eta(a)=\theta(a)+\eta$. The transformation $\theta_\eta$ is an involution if and only if $\eta\in A^-$. The transformation $\theta_\eta$ is an involution commuting with the inversion involution $(-1)\colon A\to A$ if and only if $\eta\in A^+\cap A^-$. In what follows we assume that $\eta\in A^+\cap A^-$. Then the following proposition holds (see [9]). Proposition 7.2. The real torus $(A,\theta_\eta)$ is isomorphic to the real torus $(A,\theta)$ if and only if $\eta\in A^+_0\cap A^-_0$. Writing $K_\eta=K(A_\eta)$ for the real Kummer variety $A_\eta/(-1)$, we see that Corollary 7.3 follows from Proposition 7.2. Corollary 7.3. If $\eta\in A^+_0\cap A^-_0$, then the real Kummer varieties $K(A)$ and $K(A_\eta)$ are isomorphic. Thus, if $\eta\in A^+_0\cap A^-_0$, then the real part $\mathbb{R}K(A_\eta)$ consists of topological Kummer varieties $\mathcal{K}^g$ and the number of these varieties is equal to $2^{d+1}$. The following proposition describes the structure of the real part of the Kummer variety $K(A_\eta)$ in the case when the condition $\eta\in A^+_0\cap A^-_0$ does not hold (see [9]). Proposition 7.4. If then the Kummer variety $K(A_\eta)$ has no real points. If then the real part $\mathbb{R}K(A_\eta)$ consists of topological $g$-dimensional tori and the number of these tori is equal to $2^{d-1}$. Table 1.
In Table 1, the whole rectangle represents the set $A^+\cap A^-$ of real points of order 2, and its parts represent the subsets occurring in Propositions 7.2, 7.4. The sets in Table 1 can be described in terms of the $\mathbb{F}_2$-spaces $H$ and $E^\pm$:
$$
\begin{equation*}
\begin{gathered} \, A^+_0\cap A^-_0=H,\qquad A^+_0\cap A^-=H\oplus E^-,\qquad A^-_0\cap A^+=H\oplus E^+, \\ A^+\cap A^-=H\oplus E^+\oplus E^-. \end{gathered}
\end{equation*}
\notag
$$
§ 8. The Jacobian and the Kummerian of a real hyperelliptic curveBy using some results in [32], [33], we state a number of properties of the Kummerian of a real hyperelliptic curve. We first recall an analytic definition of the Jacobian of a Riemann surface of genus $g>0$. Given such a surface $X$, we write $\Lambda$ (resp. $L$) for the homology group $H_1(X,\mathbb{Z})$ (resp. the vector space $\Omega(X)^\vee$, where $\Omega(X)$ is the space of holomorphic $1$-forms on $X$). Then integration of forms over contours in $X$ induces an embedding $\Lambda\hookrightarrow L$. By definition, the Jacobian $J(X)$ is equal to the complex torus $L/\Lambda$. If $x_0\in X$ is a fixed point, then the Abel–Jacobi map is defined by the formula
$$
\begin{equation*}
AJ_{x_0}(x)=\int_{\alpha(x)}\omega\ \operatorname{mod}\Lambda,
\end{equation*}
\notag
$$
where $\alpha(x)$ is a parametric curve on $X$ connecting the point $x_0$ to $x$, $\omega\in\Omega(X)$. Note that this map depends on the point $x_0$. But there is an Abel–Jacobi map independent of $x_0$, where $\operatorname{Div}^0 X$ is the group of divisors of degree zero on $X$. This map is defined in the following way. If then
$$
\begin{equation*}
AJ(D)=AJ_{x_0}(x_1)+\dots+AJ_{x_0}(x_k)- AJ_{x_0}(y_1)-\dots-AJ_{x_0}(y_k).
\end{equation*}
\notag
$$
The map is a group homomorphism whose kernel is the subgroup of divisors of meromorphic functions on $X$. The point $AJ(D)$ of the Jacobian is denoted by $(D)$ and is called the divisor class of $D$. Let $c\colon X\to X$ be an antiholomorphic involution (a real structure). Then there is an involution
$$
\begin{equation*}
c_*\colon H_1(X,\mathbb{Z})=\Lambda\to\Lambda=H_1(X,\mathbb{Z}).
\end{equation*}
\notag
$$
We denote it by $\theta$. It extends to an antilinear involution by the following rule. If $\ell\in\Omega(X)^\vee$, then
$$
\begin{equation*}
\theta(\ell)(\omega)=\overline{\ell(\overline{c^*(\omega)})},
\end{equation*}
\notag
$$
where the bar stands for complex conjugation. Hence we obtain a real structure on the Jacobian, In what follows we assume that the curve $X$ has real points. If the point $x_0$ is real, then the Abel–Jacobi map is real. The corresponding map of real parts, possesses the following properties (see [32], [33]).Proposition 8.1. 1) Let $k=k(J)$ be the number of components (ovals) of $\mathbb{R}X$. Then $d(J)=k-1$. 2) The map $\langle\mathbb{R}X\rangle\to\langle\mathbb{R}J\rangle$ between the sets of connected components is injective. 3) The Lie group $\mathbb{R}J$ is generated by the set $AJ_{x_0}(\mathbb{R}X)$. We proceed to consider the Jacobian of a real hyperelliptic curve. Such a curve $W$ may be regarded as a double covering of $\overline{\mathbb{C}}$ branched at points $a_1,\dots,a_{2g+2}$, where $a_1,\dots,a_{2r}$, $r>1$, is an increasing sequence of real numbers while every pair $\{a_{2r+1},a_{2r+2}\}$, $\dots$, $\{a_{2g+1},a_{2g+2}\}$ is a pair of complex-conjugate numbers and the imaginary parts of $a_{2r+2},a_{2r+4},\dots,a_{2g+2}$ are positive (see Fig. 1). We also assume that the preimage of the point $\infty\in\overline{\mathbb{C}}$ under the projection $\pi\colon W\to\overline{\mathbb{C}}$ consists of two real points. The ramification points of $W$ are projected to $a_1,\dots,a_{2g+2}$ and are denoted by $w_1,\dots,w_{2g+2}$ respectively. We connect the pairs of points $\{a_{2r},a_{2r+2}\}$, $\dots$, $\{a_{2g},a_{2g+2}\}$ by disjoint arcs $C_1,\dots,C_{g-r+1}$. Let $C'_1,\dots,C'_{g-r+1}$ be the arcs obtained from $C_1,\dots,C_{g-r+1}$ by complex conjugation (see Fig. 1). We write
$$
\begin{equation*}
\mathcal{O}_1,\ \mathcal{O}_2,\ \dots,\ \mathcal{O}_{2r-3},\ \mathcal{O}_{2r-2}
\end{equation*}
\notag
$$
for the contours on $W$ which are preimages of the intervals
$$
\begin{equation*}
[a_1,a_2],\ [a_2,a_3],\ \dots,\ [a_{2r-3},a_{2r-2}],\ [a_{2r-2},a_{2r-1}]
\end{equation*}
\notag
$$
under the projection $\pi\colon W\to\overline{\mathbb{C}}$. Fix an orientation of these contours. We write
$$
\begin{equation*}
\mathcal{O}_{2r},\ \mathcal{O}_{2r+2},\ \dots,\ \mathcal{O}_{2g-2},\ \mathcal{O}_{2g}
\end{equation*}
\notag
$$
for the contours on $W$ which are preimages of the arcs under the projection $\pi\colon W\to\overline{\mathbb{C}}$. Fix an orientation of all these contours. We write
$$
\begin{equation*}
\mathcal{O}_{2r-1},\ \mathcal{O}_{2r-3},\ \dots,\ \mathcal{O}_{2g-3},\ \mathcal{O}_{2g-1}
\end{equation*}
\notag
$$
for the contours on $W$ which are preimages of the arcs under the projection $\pi\colon W\to\overline{\mathbb{C}}$. The orientation of these contours is induced by the orientation of the contours
$$
\begin{equation*}
\mathcal{O}_{2r},\ \mathcal{O}_{2r-2},\ \dots,\ \mathcal{O}_{2g-2},\ \mathcal{O}_{2g}
\end{equation*}
\notag
$$
by means of the involution $c\colon W\to W$. Let $\lambda_1,\dots,\lambda_{2g}\in H_1(W,\mathbb{Z})$ be the homology classes determined by the oriented contours $\mathcal{O}_1,\dots,\mathcal{O}_{2g}$. We easily see that $\lambda_1,\dots,\lambda_{2g}$ is a basis of the lattice $\Lambda=H_1(W,\mathbb{Z})$ satisfying the conditions in § 7, where $d=r-1$. The curve $W$ is given by the equation Therefore any form $\omega\in\Omega(W)$ can be represented as where $p(t)$ is a polynomial of degree $\leqslant g-1$.Each oriented contour $\mathcal{O}_i$, $i=1,\dots,2g$, contains two ramification points. We denote them by $w_{\alpha(i)}$, $w_{\beta(i)}$. The points $w_{\alpha(i)}$, $w_{\beta(i)}$ divide the contour $\mathcal{O}_i$ into two oriented arcs $\mathcal{O}_i^\pm$. It follows from (8.1) that
$$
\begin{equation*}
\int_{\mathcal{O}_i^+}\omega=\int_{\mathcal{O}_i^-}\omega.
\end{equation*}
\notag
$$
Hence the Abel–Jacobi map sends the divisor $w_{\alpha(i)}-w_{\beta(i)}$ to the point
$$
\begin{equation*}
\frac{1}{2}\lambda_i\ \operatorname{mod}\Lambda\in J(W).
\end{equation*}
\notag
$$
The following Proposition 8.2 is a corollary of this fact. Proposition 8.2. The subspaces $E^+$, $E^-$, $H$ of the space ${_2}J^\theta$ are generated (respectively) by the divisor classes Using Proposition 7.1, we obtain Corollary 8.3. Corollary 8.3. If $r>0$, then the real part of the Kummerian $K(W)$ is homeomorphic to the topological space $\mathcal{K}^g_{r-1}$. § 9. Proof of Theorem 0.69.1. Sufficiency of the conditions of the theorem in the case when $r>0$Suppose that $B\,{\in}\,\mathcal{BQ}^{(r)}$, $r>0$, $W=W(B)$ is the corresponding hyperelliptic curve, $F=F(B)$ is the Fano variety, and they are real. Consider the family of Kummer varieties $K_w=F/i_w$ for such a biquadric. It follows from the definition of the involution $i_w\colon F\to F$ that it is a real map if and only if the point $w\in W$ is real. Hence the Kummer variety $K_w=F/i_w$ is real if $w\in\mathbb{R}W$. We shall prove the following lemma about such varieties. Lemma 9.1. The real Kummer varieties $K_w$, $K(W)$ are isomorphic if and only if $K_w$ has real singular points. Proof. It suffices to prove the following assertion. If $K_w$ has real points, then $K_w$ is isomorphic to $K(W)$.
The biquadric $B$ is given by an equation
$$
\begin{equation}
a_1v_1^2+\dots+a_{2r}v_{2r}^2+a_{2r+1}v_{2r+1}^2+\dots+a_nv_n^2=0,
\end{equation}
\tag{9.1}
$$
where $r>0$, the system of coordinates $(v_1,\dots,v_n)$ belongs to $\mathcal{SK}^{(r)}$, $a_1,\dots,a_{2r}$ are pairwise distinct real numbers, every pair of coefficients
$$
\begin{equation*}
\{a_{2r+1},a_{2r+2}\},\quad \dots,\quad \{a_{n-1},a_n\}
\end{equation*}
\notag
$$
is a pair of complex-conjugate numbers and these pairs are pairwise distinct. Let $w_1,\dots,w_{n}$ be the ramification points of the covering $\pi\colon W(B)\to L(B)$, which are projected to the points respectively. Since $r>0$, there are two real ramification points of $W$ on the oval $\mathcal{O}$ of the curve $\mathbb{R}W$ such that $\mathcal{O}$ contains $w$. We can assume that these points are equal to $w_1$, $w_2$.
If $w'\in\mathcal{O}$, then the topological type of the real part of the Kummer variety $K_{w'}$ is independent of the point $w'$. Therefore, for every point $w'\in\mathcal{O}$, the Kummer variety $K_{w'}$ has real singular points. We claim that the real variety $K_{w_1}$ is isomorphic to the real variety $K(W)$. Indeed, the set $F^{(w_1)}$ contains real points since the variety $K_{w_1}$ has real singular points. We denote one of the real points in this set by $o$ and take it for the zero element of $F$. Then $F$ becomes a real Abelian group variety. Consider the special embedding $\varphi_o\colon W\to F$ as in § 6.3. Since the point $o$ is real, the map $\varphi_o$ is real. Then the group isomorphism of Abelian varieties (see § 6.3) is real. It induces an isomorphism of Kummer varietiesWe now claim that the real varieties $K_{w_1}$ and $K_w$, $w\in\mathcal{O}$, are isomorphic. Indeed, since (see Proposition 6.1), the involution $i_w$ is obtained from the involution $i_0$ by translation by the real point $i_w(o)$. Hence the translation by $2i_w(o)$ induces an isomorphism of the real Kummer varieties $K_{w_1}$ and $K_w$. $\Box$Lemma 9.2. Suppose that the biquadric $B$ is given by an equation (9.1), where $a_1,\dots,a_{2r}$ is an increasing sequence. Then the real Kummer variety $K(B)$ has real singular points. Proof. We write $\mathcal{O}\subset\mathbb{R}W$ for the oval containing the real points $w_1,w_{2r}\in\mathbb{R}W$ (Fig. 2).
On Fig. 2, we write $\pm\infty\in\mathcal{O}$ for the preimages of the point $\infty\in\overline{\mathbb{C}}$ under the projection $\pi\colon W\to\overline{\mathbb{C}}$. It suffices to show that the real Kummer variety $K_{w_1}$ has real singular points. By a description given in [13], the involution $i_{w_1}\colon F\to F$ is induced by multiplication of the coordinate $v_1$ by $-1$. Therefore the set of points $F^{(w_1)}$ consists of $(g-1)$-planes on the biquadric $B'$ given by the following system of equations in $\mathbb{P}^{2g}$:
$$
\begin{equation*}
\begin{cases} v_2^2+\dots+v_n^2=0, \\ a_2v_2^2+\dots+a_{2r}v_{2r}^2+a_{2r+1}v_{2r+1}^2+\dots+a_nv_n^2=0. \end{cases}
\end{equation*}
\notag
$$
The index function $I_{B'}$ satisfies the inequalities It follows from these inequalities (see [33]) that there are real $(g-1)$-planes on the biquadric $B'$. Hence the set $F^{(w_1)}$ contains real points. Therefore the Kummer variety $K_{w_1}$ has real singular points. $\Box$ When $B$ is a real biquadric belonging to the component $|0,1|^{(r)}$, it follows from Proposition 4.3 that $B$ can be given by an equation (9.1) with an increasing sequence $a_1,\dots,a_{2r}$. Hence Lemmas 9.1, 9.2 yield the following assertion. If $B$ is a real biquadric belonging to the component $|0,1|^{(r)}$, then the real Kummer variety $K(B)$ is isomorphic to the Kummerian $K(W)$, where $W=W(B)$. This proves that the conditions of Theorem 0.6 are sufficient for the real Kummer variety $K(B)$ to be isomorphic to the Kummerian $K(W)$. 9.2. Necessity of the conditions of the theorem in the case when $r>0$We need to prove the following assertion. If $B\in\mathcal{BQ}^{(r)}$, $r>0$, is a biquadric given by an equation (9.1) such that the real varieties $K(B)$ and $K(W)$ are isomorphic, then $B$ belongs to the component $|0,1|^{(r)}$. This assertion holds for $r=1$ because $\mathcal{BQ}^{(1)}$ consists of a single component $|0,1|$,. Therefore we assume that $r>1$. Since the real Kummer variety $K(B)$ has real singular points, the Fano variety $F=F(B)$ has real points. It follows (see [34]) that the index function satisfies the double inequality But (9.2) is insufficient for proving the assertion above. We need further properties of the index function $I_{B}$.Let $a_k$ (resp. $a_s$) be the smallest (resp. the largest) of the coefficients $a_1,\dots,a_{2r}$. Then the points $w_k$, $w_s$ lie on the oval $\mathcal{O}\subset\mathbb{R}W$ that contains the points $\pm\infty$. The corresponding picture is similar to Fig. 2 with $w_1$ and $w_{2r}$ replaced by $w_k$ and $w_s$ respectively. It can also be assumed that $a_k<0$ and $a_s>0$. In what follows we always assume that these inequalities hold. The pencil of quadrics $L(B)$ is given by the equation Fig. 3 shows the circle $\lambda^2+\mu^2=1$. The tangent line above this circle is the axis $Ot$, where $t=\lambda/\mu$. The lines
$$
\begin{equation*}
\{\lambda+a_1\mu=0\},\quad \dots,\quad \{\lambda+a_{2r}\mu=0\}
\end{equation*}
\notag
$$
are also shown, and the thick lines are $\{\lambda+a_k\mu=0\}$ and $\{\lambda+a_s\mu=0\}$. The numbers inside the circle are the values of the index function $I_B$ in the corresponding sectors. An explanation of why does the index function take these values will be given below. For now, we only note that $I_B$ is indeed equal to $g+1$ in the right and left sectors bounded by the lines $\{\lambda+a_k\mu=0\}$, $\{\lambda+a_s\mu=0\}$. We proceed to explain the other values of the index function $I_B$. The number $k$ may be odd or even. We first consider the case when $k$ is odd. The section of the biquadric $B$ by the hyperplane $\{v_k=0\}$ is a biquadric of dimension $2g-2$. We denote it by $B'$. The biquadric $B'$ is given by the system of equations
$$
\begin{equation*}
\begin{cases} v_1^2+\dots+v_{k-1}^2+v_{k+1}^2+\dots+v_n^2=0, \\ a_1v_1^2+\dots+a_{k-1}v_{k-1}^2+a_{k+1}v_{k+1}^2+\dots+a_nv_n^2=0. \end{cases}
\end{equation*}
\notag
$$
Some values of the index function $I_{B'}$ are shown at Fig. 3 outside the circle. An explanation of why does $I_{B'}$ take these values will be given below. For now, we only note that below the line $\{\lambda+a_k\mu=0\}$ these values are one fewer than the corresponding values of $I_{B}$. Indeed, since the coordinate $v_k$ is real and the number $a_k$ is negative, the values of $I_B$ below the line $\{\lambda+a_k\mu=0\}$ exceed the values of $I_{B'}$ by one. In particular, the values of $I_{B'}$ in the right sector between the lines $\{\lambda+a_k\mu=0\}$, $\{\lambda+a_s\mu=0\}$ are indeed equal to $g$. The values of the index functions $I_B$, $I_{B'}$ above the line $\{\lambda+a_k\mu=0\}$ coincide. Below, we shall prove that $I_{B'}$ takes only the values $g,g+1$. Since they must alternate under each passage through the lines $\{\lambda+a_i\mu=0\}$, $i\neq k$, the assertion on the values of $I_B$ follows. In particular, we see that the index code of $B$ is of the form We proceed to prove the double inequality It follows from Lemma 9.1 that the Kummer variety $K(B)$ has real singular points. Hence the Kummer variety $K_{w_k}=F/i_{w_k}$ has real singular points. Therefore the involution $i_{w_k}\colon F\to F$ has real fixed points. But this involution is induced by the transformation of $B$ that multiplies the coordinate $v_k$ by $-1$ (see [13]). Hence the biquadric $B'$ contains real $(g-1)$-planes. But this fact is equivalent to the double inequality (9.3). As a result, we see that the index code of $B$ is of the form Applying Proposition 4.3, we deduce that $B$ belongs to the component $|0,1|^{(r)}$, as required.Thus we have proved that $B$ belongs to $|0,1|^{(r)}$ in the case when $k$ is odd. The proof in the case of even $k$ is similar. We mention only the differences. The coordinate $v_k$ is imaginary in this case. Hence the values of the index functions $I_B$ and $I_{B'}$ coincide in the sectors below the line $\{\lambda+a_k\mu=0\}$. Above this line, the values of $I_{B'}$ are one fewer than the values of $I_{B}$. As a result, we see that the index code of $B$ is of the form But this index code is equivalent to the index codeThis completes the proof of Theorem 0.6 for $r>0$. 9.3. Proof of Theorem 0.6 in the case when $r=0$In this case $B$ is given by the equation where each pair of coefficients is a pair of complex-conjugate numbers and these pairs are pairwise distinct. We again write $w_1,\dots,w_{n}$ for the ramification points of the covering $\pi\colon W(B)\to L(B)$ that are projected to the points respectively. Then $c(w_1)=w_2$, $\dots$, $c(w_{n-1})=w_n$. In this case, the special embedding $\varphi_o\colon W\to F$ is not a real map but we have a commutative diagram where $\varphi_{c(o)}\colon W\to F$ is another special embedding. Note that the point $o$ belongs to the set $F^{(w_1)}$ while $c(o)$ belongs to $F^{(w_2)}$. There is a similar commutative diagram of Abel–Jacobi maps where $\eta=AJ_{w_1}(w_2)$, that is, $\eta$ is equal to the linear equivalence class of the divisor $(w_2-w_1)$. Using the commutative diagrams (9.4) and (9.5), we obtain a commutative diagramWhen $g$ is even (resp. odd), the real part of the curve $W$ consists of a single oval (resp. two ovals); see [29], [35]. Using (7.1) and Proposition 8.1, we obtain the following assertion. If $g$ is even (resp. odd), then the real part of the Jacobian $J$ consists of a single component (resp. two components). When $g$ is even, Corollary 7.3 yields that the Kummer varieties $K(J)$, $K(J_\eta)$ are isomorphic. Hence it follows from the commutative diagram (9.6) that the real Kummer varieties $K(W)$, $K(B)$ are isomorphic. We now suppose that the real part of the Jacobian $J$ consists of two components. Then the condition $\eta\in J^+_0\cap J^-_0$ does not hold, but instead we have Therefore it follows from the commutative diagram (9.6) and Proposition 7.4 that the real part of the Kummer variety $K(B)$ consists of a single topological torus. Hence the real Kummer varieties $K(W)$, $K(B)$ are not isomorphic. This completes the proof of Theorem 0.6.Remark. The following assertion was obtained in the course of the proof of Theorem 0.6. If $r=0$ and $g$ is odd, then the real part of the Kummer variety $K(B)$ consists of a single topological torus. § 10. The topology of a real Kummer varietyThe last problem solved in the present paper concerns the topological classification of real Kummer varieties arising from real biquadrics. If $B\in\mathcal{BQ}^{(r)}$, $r>0$, is a biquadric belonging to the component $|0,1|^{(r)}$, then it follows from Theorem 0.6 and Corollary 8.3 that the real part of the Kummer variety $K(B)$ is homeomorphic to the topological space $\mathcal{K}^g_{r-1}$. In a similar vein, the real part of the Kummer variety of any biquadric $B$ in $\mathcal{BQ}^{(0)}$ is homeomorphic to $\mathcal{K}^g_0$ when $g$ is even. But if $B$ belongs to $\mathcal{BQ}^{(0)}$ and $g$ is odd, then the real part of $K(B)$ consists of a single $g$-dimensional topological torus (see the remark at the end of § 9). Since $\mathcal{BQ}^{(1)}$ consists of a single component $|0,1|$, we need only to find the topological type of $\mathbb{R}K(B)$ in the case when $B\in\mathcal{BQ}^{(r)}$, $r>1$, and $B$ does not belong to the component $|0,1|^{(r)}$. In this case, it turns out that the topological space $\mathbb{R}K(B)$ has no singular points (see § 9.1) and the following assertion holds. If $\mathbb{R}K(B)$ is non-empty, then it consists of $g$-dimensional topological tori and the number of tori is $2^{r-2}$. Thus we need to know whether $\mathbb{R}K(B)$ is non-empty. We shall prove Theorem 10.1. Theorem 10.1. The real part of the Kummer variety $K(B)$ of a biquadric $B\in\mathcal{BQ}^{(r)}$, $r>1$, is non-empty if and only if one of the following conditions holds. 1) The $\mathbb{F}_2$-code of $B$ is equal to $|0,1,0,1,\dots,0,1|$. 2) There is an increasing sequence of odd integers $i_1<\dots<i_k$, $i_1\geqslant1$, $i_k\leqslant 2r-1$, different from the sequence $1,3,\dots,2r-1$ and such that the $\mathbb{F}_2$-code of $B$ is obtained from the $\mathbb{F}_2$-code by replacing zeros by ones at places $i_1,\dots,i_k$ and ones by zeros at places $i_1+ 1, \dots, i_k+1$.3) There is an increasing sequence of even integers $i_1<\dots< i_k$, $i_1\geqslant2$, $i_k\leqslant 2r-2$, such that the $\mathbb{F}_2$-code of $B$ is obtained from the $\mathbb{F}_2$-code by replacing ones by zeros at places $i_1,\dots,i_k$ and zeros by ones at places $i_1+ 1,\dots, i_k+1$.This theorem yields Corollary 10.2. Corollary 10.2. The real part of the Kummer variety $K(B)$ of a biquadric $B\in\mathcal{BQ}^{(r)}$, $r>1$, consists of $g$-dimensional topological tori if and only if the second or third condition of Theorem 10.1 holds. In this case, the number of tori is equal to $2^{r-2}$. When neither condition of Theorem 10.1 holds, the real part of $K(B)$ is empty. When we consider biquadrics belonging to a fixed component of $\mathcal{BQ}^{(r)}$, $r>1$, the topological type of $\mathbb{R}K(B)$ is independent of $B$. Therefore, to find the topological type of $\mathbb{R}K(B)$ for all biquadrics in the components of $\mathcal{BQ}^{(r)}$, it suffices to choose a sample biquadric $B$ in each component of $\mathcal{BQ}^{(r)}$ and find the topological type of the corresponding space $\mathbb{R}K(B)$. 10.1. Examples of biquadricsIn what follows we fix a biquadric $B_0\in\mathcal{BQ}^{(r)}$, $r>1$, in the component $|0,1,0,1,\dots,1,0|$. Suppose that it is given by an equation
$$
\begin{equation}
a_1v_1^2+\dots+a_{2r}v_{2r}^2+a_{2r+1}v_{2r+1}^2+\dots+a_nv_n^2=0,
\end{equation}
\tag{10.1}
$$
where $(v_1,\dots,v_n)\in\mathcal{SK}^{(r)}$, $r>1$, $a_1,\dots,a_{2r}$ is an increasing sequence of real coefficients, every pair of coefficients
$$
\begin{equation*}
\{a_{2r+1},a_{2r+2}\},\quad\dots,\quad\{a_{n-1},a_n\}
\end{equation*}
\notag
$$
is a pair of complex-conjugate numbers, all these pairs are pairwise distinct, and the imaginary parts of $a_{2r+2},a_{2r+4},\dots,a_{2g+2}$ are positive. We shall vary $B_0$ in order to obtain biquadrics $B$ in components different from For every permutation $\sigma\in\mathfrak{S}_{2r}$ let $B_\sigma\in\mathcal{BQ}^{(r)}$ be the biquadric given by the equation
$$
\begin{equation}
a_{\sigma(1)}v_1^2+\dots+a_{\sigma(2r)}v_{2r}^2+ a_{2r+1}v_{2r+1}^2+\dots+a_nv_n^2=0.
\end{equation}
\tag{10.2}
$$
The equation (10.2) coincides with the equation
$$
\begin{equation}
a_1v_{\sigma^{-1}(1)}^2+\dots+a_{2r}v_{\sigma^{-1}(2r)}^2+a_{2r+1}v_{2r+1}^2+\dots+ a_nv_n^2=0.
\end{equation}
\tag{10.3}
$$
Let $(v_1,\dots,v_{2r},v_{2r+1},\dots,v_{n})$ be a point satisfying the equation (10.1). Then the point
$$
\begin{equation*}
(v_{\sigma(1)},\dots,v_{\sigma(2r)},v_{2r+1},\dots,v_{n})
\end{equation*}
\notag
$$
satisfies the equation (10.3). Hence we obtain an isomorphism of complex biquadrics which is given by the rule
$$
\begin{equation*}
f_\sigma(v_1,\dots,v_{2r},v_{2r+1},\dots,v_{n})= (v_{\sigma(1)},\dots,v_{\sigma(2r)},v_{2r+1},\dots,v_{n}).
\end{equation*}
\notag
$$
The biquadrics $B_0$, $B_\sigma$ are real. Denoting their native real structures by
$$
\begin{equation*}
c_0\colon B_0\to B_0,\qquad c\colon B_\sigma\to B_\sigma
\end{equation*}
\notag
$$
and writing $c_\sigma\colon B_0\to B_0$ for the real structure on $B_0$ equal to the antiholomorphic involution $f_\sigma^{-1}\circ c\circ f_\sigma$, we obtain an isomorphism of real biquadrics
$$
\begin{equation*}
f_\sigma\colon (B_0,c_\sigma)\xrightarrow{\approx}(B_\sigma,c).
\end{equation*}
\notag
$$
We now compare the real structures $c_0,c_\sigma$ on the biquadric $B_0$. Recall that the coordinate functions $v_1,\dots,v_{2r}$ with odd (resp. even) numbers are real (resp. imaginary). Let be an increasing sequence of odd numbers consisting of all the numbers such that $\sigma^{-1}(i)$ is even, and let be an increasing sequence of even numbers consisting of all the numbers such that $\sigma^{-1}(j)$ is odd. Then we write $(-1_\sigma)\colon B_0\to B_0$ for the transformation that multiplies the coordinates with numbers $i_1,\dots,i_k,j_1,\dots,j_k$ by $-1$. It follows from the definitions of $c_0$, $c_\sigma$, $(-1_\sigma)$ thatSince
$$
\begin{equation*}
\operatorname{cd}(a_{\sigma(1)},\dots,a_{\sigma(2r)}) =\sigma^{-1}\cdot\operatorname{cd}(a_1,\dots,a_{2r})=\sigma^{-1}\cdot(2r,\dots,1),
\end{equation*}
\notag
$$
the following lemma holds. Lemma 10.3. The $\mathbb{F}_2$-code of $B_\sigma$ is obtained from the $\mathbb{F}_2$-code $|0,1,0,1,\dots,0,1|$ by replacing zeros by ones at places $i_1,\dots,i_k$ and ones by zeros at places $j_1,\dots,j_k$. Hence the $\mathbb{F}_2$-code of the biquadric $B_\sigma$ may be arbitrary. 10.2. The Fano variety $F(B_0)$Since the biquadric $B_0$ belongs to the component $|1,0,\dots,1,0|$, the real Fano variety $F=F(B_0)$ is isomorphic to the Jacobian $J=J(W)$, where $W=W(B_0)$. We write $w_1,\dots,w_n$ for the ramification points of the projection $\pi\colon W\to\overline{\mathbb{C}}$ enumerated in accordance with the enumeration of $a_1,\dots,a_n$. The points $w_1,\dots,w_{2r}$ are real and the pairs of points $\{w_{2r+1},w_{2r}\}$, $\dots$, $\{w_{n-1},w_{n}\}$ are complex-conjugate. Suppose that $o\in F^{(w_1)}$ is a real point. Then there is a real group isomorphism of Abelian varieties (see § 6.3 and § 9.1) The transformation $(-1_\sigma)\colon B_0\to B_0$ induces a transformation of $F$, which is denoted again by $(-1_\sigma)\colon F\to F$. Let $\eta_\sigma$ be the point of order two in $J$ equal to the divisor class Then Lemma 10.4 follows from [13].Lemma 10.4. The transformation $(-1_\sigma)\colon F\to F$ is equal to the transformation of translation by the point $\eta_\sigma$ of order two, that is, $(-1_\sigma)=t_{\eta_\sigma}$. 10.3. Completion of the proof of Theorem 10.1Since the $\mathbb{F}_2$-code of $B_\sigma$ may be arbitrary (see Lemma 10.3), every component of $\mathcal{BQ}^{(r)}$, $r>1$, contains a biquadric of the form $B_\sigma$. We first prove that if $B\in\mathcal{BQ}^{(r)}$ is a biquadric satisfying one of the conditions in Theorem 10.1, then the real part of $K(B)$ is non-empty. If the $\mathbb{F}_2$-code of $B$ is equal to $|0,1,0,1,\dots,0,1|$, then the real variety $K(B)$ is isomorphic to the real Kummerian $K(W)$, where $W=W(B)$. Hence the structure of $\mathbb{R}K(B)$ follows from § 7 and, in particular, $\mathbb{R}K(B)$ is non-empty. Suppose that there is an increasing sequence of odd numbers
$$
\begin{equation*}
i_1,\dots,i_k,\qquad i_1\geqslant1,\quad i_k\leqslant 2r-1,
\end{equation*}
\notag
$$
such that the $\mathbb{F}_2$-code of $B$ is obtained from the $\mathbb{F}_2$-code by replacing zeros by ones at places $i_1,\dots,i_k$ and ones by zeros at places $i_1+ 1,\dots, i_k+ 1$. Let $\sigma$ be the permutation (involution) that sends $i_1,\dots,i_k$ to $i_1+1,\dots,i_k+1$ respectively (and accordingly sends $i_1+1,\dots,i_k+1$ back to $i_1,\dots,i_k$) and leaves all other numbers fixed. The biquadrics $B$ and $B_\sigma$ have the same $\mathbb{F}_2$-code and, therefore, belong to the same component of $\mathcal{BQ}^{(r)}$. We can assume that this component is different from $|0,1,0,1,\dots,0,1|$. Hence the real part of $K(B)$ is non-empty if and only if the real part of $K(B_\sigma)$ is non-empty. Using (10.4), Lemma 10.4 and Proposition 7.4, we see that the real part of $K(B_\sigma)$ is non-empty if and only if
$$
\begin{equation}
\eta_\sigma\in(J^+_0\cap J^-\setminus J^+_0\cap J^-_0)\cup(J^-_0\cap J^+ \setminus J^+_0\cap J^-_0).
\end{equation}
\tag{10.5}
$$
Since
$$
\begin{equation*}
\eta_\sigma=(w_{i_1}+\dots+w_{i_k}-w_{i_1+1}-\dots-w_{i_k+1})= (w_{i_1}-w_{i_1+1})+\dots+(w_{i_k}-w_{i_k+1}),
\end{equation*}
\notag
$$
it follows from Proposition 8.2 that $\eta_\sigma\in E^-$. Hence (10.5) holds. Therefore the real parts of $K(B_\sigma)$ and $K(B)$ are non-empty. A similar argument shows that if $B\in\mathcal{BQ}^{(r)}$, $r>1$, is a biquadric satisfying the third condition of Theorem 10.1, then the real part of $K(B)$ is non-empty. This proves Theorem 10.1. 10.4. The topological type of $\mathbb{R}K(B)$ in the general caseCombining Corollary 10.2 with the assertions stated before Theorem 10.1, we obtain the following theorem. Theorem 10.5. The following assertions hold for the topological space $\mathbb{R}K(B)$, where $B\in\mathcal{BQ}^\circ$. 1) If $B\in\mathcal{BQ}^{(0)}$ and $g$ is odd, then $\mathbb{R}K(B)$ is homeomorphic to a $g$-dimensional topological torus. 2) If $B\in\mathcal{BQ}^{(0)}$ and $g$ is even, then $\mathbb{R}K(B)$ is homeomorphic to $\mathcal{K}^g_0$. 3) If $B\in|0,1|^r$, then $\mathbb{R}K(B)$ is homeomorphic to $\mathcal{K}^g_{r-1}$. 4) The real part of the Kummer variety $K(B)$ of a biquadric $B\in\mathcal{BQ}^{(r)}$, $r>1$, consists of $g$-dimensional topological tori if and only if the second or third condition of Theorem 10.1 holds. In this case, the number of tori is equal to $2^{r-2}$. When neither condition of Theorem 10.1 holds, the real part of $K(B)$ is empty. 10.5. The topological type of $\mathbb{R}K(B)$ for $n=6$When the biquadric $B\in\mathcal{BQ}^\circ$ is three-dimensional, the Kummer variety $K(B)$ is a surface. In this case we have Corollary 10.6 of Theorem 10.5. Corollary 10.6. Suppose that $n=6$ and $K\in\mathcal{KV}^\circ$. Then the following assertions hold. 1) If $K\in[\ ]\cup[01]$, then the topological surface $\mathbb{R}K$ consists of two topological two-dimensional spheres $\mathcal{S}^\pm$ which have four common points. 2) If $K\in[0101]$, then the topological surface $\mathbb{R}K$ consists of four topological two-dimensional spheres $\mathcal{S}^\pm_1$, $\mathcal{S}^\pm_2$, and every sphere $\mathcal{S}^+_i$ has two common points with every sphere $\mathcal{S}^-_j$, $i=1,2$, $j=1,2$. 3) If $K\in[010101]$, then the topological surface $\mathbb{R}K$ consists of eight topological two-dimensional spheres $\mathcal{S}^\pm_1$, $\dots$, $\mathcal{S}^\pm_4$, and every sphere $\mathcal{S}^+_i$ has a single common point with every sphere $\mathcal{S}^-_j$, $i=1,\dots,4$, $j=1,\dots,4$. 4) If $K\in[0011]$, then the topological surface $\mathbb{R}K$ is a two-dimensional topological torus. 5) If $K\in[001011]$, then the topological surface $\mathbb{R}K$ is the disjoint union of two two-dimensional topological tori. 6) If $K\in[000111]$, then $\mathbb{R}K$ is the empty set. Remark. When $n=6$, the set $\mathcal{KV}\setminus\mathcal{KV}^\circ$ consists of a single connected component of $\mathcal{KV}$ (see [10]). The Kummer surfaces in this component are referred to as non-standard in [10]. Their deviation from the standard is that the real part consists of four points. It is quite possible that analogous assertions hold for $n>6$.
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\Bibitem{Kra22} |
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