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Birationally rigid hypersurfaces with quadratic singularities of low rank A. V. Pukhlikov Department of Mathematical Sciences, University of Liverpool, Liverpool, UK
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     Introduction0.1. The statement of the main resultLet $\mathcal{P}=\mathcal{P}_{M,M+1}$ be the linear space of homogeneous polynomials of degree $M$ in $M+1$ variables, where $M\geqslant 5$, which we identify with the space of sections $H^0({\mathbb P}^M,\mathcal{O}_{{\mathbb P}^M}(M))$. The projectivization of the space $\mathcal{P}$ parameterizes hypersurfaces of degree $M$ in ${\mathbb P}^M$. If for $f\in\mathcal{P}\setminus\{0\}$ the hypersurface $F(f)=\{f=0\}$ is irreducible, reduced, factorial and has terminal singularities, then $F(f)\subset{\mathbb P}^M$ is a primitive Fano variety:
$$
\begin{equation*}
\operatorname{Pic}F(f)={\mathbb Z}H\qquad\text{and} \qquad K_{F(f)}=-H,
\end{equation*}
\notag
$$
where $H$ is the class of a hyperplane section. Let $\mathcal{F}_{\rm srigid}\subset\mathcal{P}$ be the subset consisting of the polynomials $f\in\mathcal{P}\setminus\{0\}$ such that the hypersurface $F(f)$ satisfies all properties listed above and, moreover, is a birationally superrigid variety, that is, for every linear system $\Sigma\subset|nH|$ with no fixed components, where $n\geqslant 1$, and for a general divisor $D\in\Sigma$ the pair $(F(f),\frac{1}{n}D)$ is canonical. Because the property of being birationally (super)rigid is important for problems in higher-dimensional birational geometry (a birationally superrigid variety $F(f)$ cannot be fibred into rationally connected varieties over a positive-dimensional base, is not birational to the total space of any Mori fibre space over a positive-dimensional base, any birational map from this variety onto a Fano variety of the same dimension with terminal ${\mathbb Q}$-factorial singularities is an isomorphism), we obtain the natural problem of describing the subset $\mathcal{F}_{\rm srigid}$ as precisely as possible and, in particular, of estimating the codimension of the complement $\operatorname{codim}((\mathcal{P}\setminus\mathcal{F}_{\rm srigid})\subset\mathcal{P})$. In [1] it was shown that $\mathcal{F}_{\rm srigid}$ contains the equations of all hypersurfaces with at most quadratic singularities of rank $\geqslant 5$ that satisfy the pointwise regularity conditions, which implies the estimate
$$
\begin{equation*}
\operatorname{codim}((\mathcal P\setminus\mathcal F_{\mathrm{srigid}})\subset\mathcal P)\geqslant \binom{M-3}{2}+1.
\end{equation*}
\notag
$$
The aim of this paper is to improve this result: we show that quadratic singular points of rank $4$ and $3$ satisfying certain additional conditions of general position, can also be allowed, whereas the pointwise regularity conditions can slightly be relaxed, which implies a stronger estimate for the codimension of the nonrigid hypersurfaces than the one shown in [1]. Let us give the precise statements. Let $f\in\mathcal{P}\setminus\{0\}$, and let $F=F(f)$ be the corresponding hypersurface. For a point $o\in F$ take an arbitrary system of affine coordinates $z_1,\dots,z_M$ on an affine chart ${\mathbb A}^M\subset{\mathbb P}^M$ containing this point, where $o=(0,\dots,0)$, and write the corresponding affine polynomial (which for convenience we denote by the same symbol $f$) as the sum where each $q_i$ is homogeneous of degree $i$. Obviously, $o\in\operatorname{Sing}F$ if and only if $q_1\equiv 0$. We say that that a singular point $o\in F$ is quadratic of rank $a\geqslant 1$ if ${\operatorname{rk}q_2=a}$.Assume that the point $o\in F$ is a quadratic singularity of rank $3$; in this case we may assume that $q_2=z^2_1+z^2_2+z^2_3$. We say that the point $o$ satisfies condition (G) if the singular locus of the cubic hypersurface in the projective space ${\mathbb P}^{M-4}$ with homogeneous coordinates $(z_4:\dotsb:z_M)$ is zero-dimensional or empty (in particular, $q_3(0,0,0,z_4,\dots,z_M)\not\equiv 0$) and, moreover, the homogeneous polynomial
$$
\begin{equation*}
4q_4(0,0,0,z_4,\dots,z_M)-\sum^3_{i=1}\frac{\partial q_3}{\partial z_i}(0,0,0,z_4,\dots,z_M)^2
\end{equation*}
\notag
$$
of degree $4$ does not vanish at any singular point of this hypersurface. Theorem 0.1. Assume that every point of the hypersurface $F$ is either nonsingular or is a quadratic singularity of rank $\geqslant 3$, every quadratic singularity of rank $3$ satisfies condition (G) and the inequality holds. Then $F$ is an irreducible reduced factorial variety with terminal singularities. (Of course, irreducibility, reducedness and factoriality follow from the inequality for the codimension of the subset $\operatorname{Sing}F$.) Still using the coordinate notation introduced above, note that if $o\in F$ is a nonsingular point, then the tangent hyperplane $T_oF$ is given by the equation $q_1=0$. We say that a nonsingular point $o\in F$ is regular if for $M\geqslant 6$ (R1) the polynomials form a regular sequence, that is, the system of equations has a four-dimensional set of common zeros in the hyperplane $T_oF$.We say that a quadratic singularity $o\in F$ of rank $a\geqslant 7$ (for $M\geqslant 7$) is regular, if (R2) the polynomials form a regular sequence, that is, the set of their common zeros is of dimension $5$.Finally, if $o\in F$ is a quadratic singularity of rank $\in\{3,4,5,6\}$, then we say that it is regular if (R3) the polynomials form a regular sequence, that, the set of their common zeros is one-dimensional.Now let us define a subset $\mathcal{F}\subset\mathcal{P}$ in the following way: $f\in\mathcal{F}$ if and only if the hypersurface $F=F(f)$ satisfies the assumptions of Theorem 0.1, and for $M\geqslant 6$ the inequality
$$
\begin{equation*}
\operatorname{codim}(\operatorname{Sing}F\subset F)\geqslant 5
\end{equation*}
\notag
$$
holds, while for $M=5$ the hypersurface $F$ does not contain two-dimensional planes and no section of this hypersurface by a three-dimensional subspace in ${\mathbb P}^5$ contains a line consisting of singular points of this section, and (for every $M\geqslant 5$) each point of $F$ is regular in the sense of the corresponding condition (R1), (R2) or (R3). Theorem 0.2. For $f\in\mathcal{F}$ the hypersurface $F=F(f)$ is a birationally superrigid Fano variety. In particular, $F$ cannot be fibred by a rational map into rationally connected varieties over a positive-dimensional base, $F$ is not birational to the total space of any Mori fibre space over a positive-dimensional base and every birational map $\chi\colon F\dashrightarrow F'$, where $F'$ is a Fano variety with terminal ${\mathbb Q}$-factorial singularities and the Picard number $1$, is a biregular isomorphism. Let us define a function $\gamma\colon \{M\in {\mathbb Z}\mid M\geqslant 5\}\to {\mathbb Z}$ in the following way:
$$
\begin{equation*}
\begin{array}{|c|c|c|c|c|} \hline M & 5 & 6 & 7 & \geqslant 8 \\ \hline \gamma(M) & 6 & 9 & 15 & \binom{M-1}{2}+1\vphantom{\rule{0pt}{12pt}} \\ \hline \end{array}
\end{equation*}
\notag
$$
Theorem 0.3. The codimension of the complement $\mathcal{P}\setminus\mathcal{F}$ in the space $\mathcal{P}$ is at least $\gamma(M)$. 0.2. The structure of the paperIn § 1 we prove Theorem 0.1. In addition, we describe some local conditions at a quadratic point $o\in F$ of rank $4$ that ensure the inequality $\operatorname{codim}_o(\operatorname{Sing} F\subset F)\geqslant 4$ (the symbol $\operatorname{codim}_o$ denotes the codimension in a neighbourhood of the point $o$). We show that if the condition (G) is satisfied, then there are finitely many quadratic singularities of rank $3$ and, blowing up this finite set, we obtain a variety with at most quadratic points of rank $\geqslant 4$, where the singular locus of this variety is of codimension $\geqslant 4$. In § 2 we prove Theorem 0.2. In order to exclude maximal singularities over nonsingular points we use the $8n^2$-inequality (see [2] and [3]), in the case of quadratic singularities of rank $\geqslant 7$ we use the generalized $4n^2$-inequality [4], in the case of quadratic singularities of rank $a\in\{3,4,5,6\}$ we use the traditional $4n^2$-inequality for quadratic points established in [1] with a correction made in [5]. In order to obtain a contradiction, we use the technique of hypertangent divisors based on the regularity conditions (R1), (R2) and (R3), respectively. In § 3 we show Theorem 0.3. Here we use the technique developed in [5]–[7]. 0.3. General commentsThe present paper belongs to the series of papers on effective birational rigidity, started by [1]: given a family of Fano varieties, we do not only prove the birational rigidity of a variety of general position but also give an explicit estimate for the codimension of the complement to the set of birationally rigid varieties in this family. One of the applications of such results is the possibility to construct fibrations into Fano varieties over a positive-dimensional base, each fibre of which is a birationally rigid variety: see the survey [8]. The approach to the study of geometric objects ranging in some family, when objects of general position (in some precise sense, for instance, objects with no singularities) are considered first, and after that more and more special subfamilies are studied, is sometimes called ‘Poincaré’s strategy’. If we have a family of Fano varieties, defined by means of a particular geometric construction, then it is natural to study the problem of their birational rigidity in exactly this way, allowing more and more complicated singularities. For (irreducible reduced) factorial hypersurfaces of degree $M$ in ${\mathbb P}^M$ the final goal is to determine precisely the boundaries of the set of (super)rigid varieties, in the sense of how ‘bad’ the singularities can be while the variety is still birationally rigid; for instance, in [9] it was shown that the multiplicity of the unique singular point of general position can be as high as $M-2$ (and the conditions of general position were relaxed and the proof simplified in [10]), whereas a hypersurface of degree $M$ with a point of multiplicity $M-1$ is obviously rational and cannot be birationally rigid. As an example of such an approach to the birational geometry of three-dimensional fibrations into del Pezzo surfaces, when varieties with progressively more complicated singularities are considered, we refer to the papers [11]–[13]. AcknowledgementsThe author is grateful to the members of the Departments of Algebraic Geometry and Algebra of the Steklov Mathematical Institute for their interest in his work, and to his colleagues in the Algebraic Geometry Research Group at the University of Liverpool for general support. § 1. Factorial terminal singularitiesIn this section we prove Theorem 0.1. In § 1.1 we consider simple general facts about quadratic singularities, in § 1.2 we look at what happens to singularities when we blow up a quadratic point of a given rank. In § 1.3, on this basis we prove Theorem 0.1. 1.1. Quadratic singularitiesRecall (see [1] and [14]) that a point $o\in\mathcal{X}$ of an irreducible algebraic variety $\mathcal{X}$ is a quadratic singularity of rank $r\geqslant 1$, if in some neighbourhood of this point $\mathcal{X}$ is realized as a subvariety of a nonsingular $N=(\operatorname{dim}\mathcal{X}+1$)-dimensional variety $\mathcal{Y}\ni o$, and for some system $(u_1,\dots,u_N)$ of local parameters on $\mathcal{Y}$ at the point $o$ the subvariety $\mathcal{X}$ is a hypersurface given by an equation
$$
\begin{equation*}
g\in\mathcal O_{o,\mathcal Y}\subset{\mathbb C}[[u_1,\dots,u_N]],
\end{equation*}
\notag
$$
which is represented by a formal power series where $g_j(u_1,\dots,u_N)$ is a homogeneous polynomial of degree $j$ and $\operatorname{rk}g_2=r$. The following obvious fact is true. Proposition 1.1. Assume that $o\in\mathcal{X}$ is a quadratic singularity of rank $r$. Then every point $p\in\mathcal{X}$ in a neighbourhood of $o$ is either nonsingular, or a quadratic singularity of rank $\geqslant r$. Let $\mathcal{Y}^+\to\mathcal{Y}$ be the blow-up of the point $o$ with exceptional divisor $E_\mathcal{Y}\cong{\mathbb P}^{N-1}$ and $\mathcal{X}^+\subset\mathcal{Y}^+$ the strict transform of the hypersurface $\mathcal{X}$ on $\mathcal{Y}^+$, so that $\mathcal{X}^+\to\mathcal{X}$ is the blow-up of the point $o$ on $\mathcal{X}$ with exceptional divisor $E_\mathcal{Y}|_{\mathcal{X}^+}=E_\mathcal{X}$. Therefore, $E_\mathcal{X}$ is a quadric of rank $r$ in the projective space $E_\mathcal{Y}$. The following fact is also obvious. Proposition 1.2. The equality $\operatorname{codim}(\operatorname{Sing}E_\mathcal{X}\subset\mathcal{X})=r$ and the inequality hold. From here it follows immediately in view of Grothendieck’s theorem on parafactoriality (see [15] and [16]) that a variety with at most quadratic points of rank $\geqslant 5$ is factorial (which was used in [1]). Quadratic singularities have the following useful property of being stable with respect to blow-ups (see [1] or [17], § 1.7). Proposition 1.3. Assume that $\mathcal{X}$ has at most quadratic singularities of rank ${\geqslant a\geqslant 3}$ and $B\subset\mathcal{X}$ is an irreducible subvariety of codimension $\geqslant 2$. Then there is an open subset $U\subset\mathcal{X}$ such that $U\cap B\neq\varnothing$, the subvariety $U\subset B$ is nonsingular and its blow-up produces a quasi-projective variety $U_B$ with at most quadratic singularities of rank ${\geqslant a}$. (For a proof, which is based on simple coordinate computations, see [17], § 1.7.) The arguments given above imply that if $\mathcal{X}$ is a variety with at most quadratic singularities of rank $\geqslant 5$, then $\mathcal{X}$ is a factorial variety with terminal singularities. In the notation of Proposition 1.3 for $a=5$, for $\operatorname{codim}B\in\{2,3\}$ we may assume that $U$ is nonsingular (since $B\not\subset\operatorname{Sing}\mathcal{X}$), so that the discrepancy of the exceptional divisor $E_B$ of the blow-up $\sigma_B$ is $\operatorname{codim}B-1$, and for $\operatorname{codim}B\geqslant 4$ the inequality $a(E_B,U)\geqslant\operatorname{codim}B-2$ holds (where the equality holds precisely when $B\subset\operatorname{Sing}\mathcal{X}$). However, in the present paper we allow quadratic singularities of rank $4$ and $3$. 1.2. The blow-up of a quadratic pointLet $o\in\mathcal{X}$ be a quadratic singularity of rank $a\geqslant 3$. In the notation of § 1.1 the local parameters $u_1,\dots,u_N$ can be identified with homogeneous coordinates on $E_\mathcal{Y}$, so that $E_\mathcal{X}$ is a quadric with the equation $g_2=0$. We may assume that $g_2=u^2_1+\dots+u^2_a$, so that $\operatorname{Sing}E_\mathcal{X}=\{u_1=\dots=u_a=0\}$. Set $Q=\{g_3|_{\operatorname{Sing}E_\mathcal{X}}=0\}$, that is, $Q$ is the hypersurface $g_3(0,\dots,0,u_{a+1},\dots,u_N)=0$ in the projective space ${\mathbb P}^{N-a-1}$ with homogeneous coordinates $(u_{a+1}:\cdots:u_N)$. (Of course, the linear subspace $\operatorname{Sing}E_\mathcal{X}$ and the cubic hypersurface $Q\subset\operatorname{Sing}E_\mathcal{X}$ do not depend on the system of local parameters at the point $o$.) Also set
$$
\begin{equation*}
h(u_{a+1},\dots,u_N)=4g_4(0,\dots,0,u_{a+1},\dots,u_N)- \sum^a_{i=1}\frac{\partial g_3}{\partial u_i}(0,\dots,0,u_{a+1},\dots,u_N)^2.
\end{equation*}
\notag
$$
This is a homogeneous polynomial of degree $4$ on the subspace $\operatorname{Sing}E_\mathcal{X}\subset E_\mathcal{Y}$. Proposition 1.4. (i) The following equality holds: (ii) A point $p\in Q\setminus\operatorname{Sing}Q$ is a quadratic singularity of rank $a+2$. (iii) A point $p\in\operatorname{Sing}Q$ is a quadratic singularity of rank $a+1$ if $h(p)\neq 0$ and of rank $a$ otherwise. (Note that if $g_3|_{\operatorname{Sing}E_\mathcal{X}}\equiv 0$, that is, $Q=\operatorname{Sing}E_\mathcal{X}$, then $Q=\operatorname{Sing}Q$: the singularities of $Q$ are common zeros of the partial derivatives of the cubic polynomial $g_3(0,\dots,0,u_{a+1},\dots,u_N$).) Proof. This is established by means of obvious local computations. Let $p\in\operatorname{Sing}E_\mathcal{X}$ be an arbitrary point. We may assume that
$$
\begin{equation*}
p=(\underbrace{0,\dots,0}_a,1,\underbrace{0,\dots,0}_{N-a-1}),
\end{equation*}
\notag
$$
so that a system of local parameters $w_1,\dots,w_N$ on $\mathcal{Y}^+$ at the point $p$ is related to the original system by the equalities for $i\neq a+1$. Now claim (i) is obvious. If $p\in Q$, then the first (quadratic) component in the presentation of the local equation of the variety $\mathcal{X}^+$ at the point $p$ has the form where (the only one occupies the $(a+1)$st position). Computing the rank of this quadratic form we obtain claims (ii) and (iii).
The proposition is proved. If $a=4$, then, since
$$
\begin{equation*}
E_\mathcal Y\cap\operatorname{Sing}\mathcal X^+\subset\operatorname{Sing}E_\mathcal X,
\end{equation*}
\notag
$$
we obtain the following claim. Proposition 1.5. If $o\in\mathcal{X}$ is a quadratic singularity of rank $4$ and the inequality $\operatorname{codim}(\operatorname{Sing}\mathcal{X}\subset\mathcal{X})\geqslant 4$ holds, then a similar inequality is also true for $\mathcal{X}^+$. Moreover, Proposition 1.4 implies the following fact. Proposition 1.6. If $o\in\mathcal{X}$ is a quadratic singularity of rank $4$ and either $g_3|_{\operatorname{Sing} E_\mathcal{X}}\not\equiv 0$, or $g_3|_{\operatorname{Sing} E_\mathcal{X}}\equiv 0$ but $h|_{\operatorname{Sing}E_\mathcal{X}}\not\equiv 0$, then the inequality holds. Proof. Indeed, if $Q\neq\operatorname{Sing}E_\mathcal{X}$, then
$$
\begin{equation*}
\operatorname{codim}((E_\mathcal Y\cap\operatorname{Sing}\mathcal X^+)\subset E_\mathcal X)=4,
\end{equation*}
\notag
$$
which implies the required inequality. If $Q=\operatorname{Sing}E_\mathcal{X}$, then at the point of general position $p\in Q$ the rank of the quadratic singularity of the variety $\mathcal{X}^+$ is equal to $5$. Therefore, in a neighbourhood of the point $o$ the rank of every singularity of the variety $\mathcal{X}$ is at least $4$, and the closed set of the points of rank $4$ is of codimension ${\geqslant 4}$. Now we apply Proposition 1.2 and complete the proof. 1.3. Singularities of rank $3$Let us prove Theorem 0.1. Let $o\in F$ be a quadratic point of rank $3$, ${\mathbb P}^+\to{\mathbb P}^M$ the blow-up of this point with exceptional divisor $E_{\mathbb P}\cong{\mathbb P}^{M-1}$ and $F^+\subset{\mathbb P}^+$ the strict transform of $F$, so that $E_F=F^+\cap E_{\mathbb P}$ is a quadric hypersurface of rank $3$. In the notation of § 0.1 set
$$
\begin{equation*}
Q=\{q_3(0,0,0,z_4,\dots,z_M)=0\}=\{q_3|_{\operatorname{Sing} E_F}=0\}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
h(z_4,\dots,z_M)=4q_4(0,0,0,z_4,\dots,z_M)-\sum^3_{i=1}\frac{\partial q_3}{\partial z_i}(0,0,0,z_4,\dots,z_M)^2;
\end{equation*}
\notag
$$
this notation agrees with § 1.2. Now Proposition 1.4 implies that an arbitrary point $p\in E_F$ is either nonsingular on $F^+$ (this occurs precisely when $p\notin Q$), or is a quadratic singularity of rank $5$ (when $p\in Q\setminus\operatorname{Sing}Q$), or a quadratic singularity of rank $4$ (when $p\in\operatorname{Sing}Q$). In particular, in a neighbourhood of the point $o$ there are no other quadratic singularities of rank $3$. We have shown the following fact. Proposition 1.7. In the assumptions of Theorem 0.1 the set of quadratic singularities of the hypersurface $F$ of rank $3$ is either empty or finite. In the second case, blowing up this finite set of points we obtain a variety that has, at worst, quadratic singularities of rank $\geqslant 4$, and its singular locus is of codimension $4$. Now let $E$ be a prime exceptional divisor over $F$ and $B$ its centre on $F$, and let $o\in B$ be a point of general position on $B$. If $o$ is not a quadratic singularity of rank $3$, then Propositions 1.3 and 1.5 imply that $a(E,F)\geqslant 1$. Assume that $B=o$ is a quadratic singularity of rank $3$. Then either $E=E_F$ and $a(E,F)=M-3\geqslant 2$, or the centre of $E$ on $F^+$ is an irreducible subvariety $B_+\subset E_F$ of codimension $\geqslant 2$, so that by what we said above about the singularities of the variety $F^+$ we obtain This proves that the singularities of the variety $F$ are terminal. The proof of Theorem 0.1 is complete.§ 2. Birational superrigidityIn this section we prove Theorem 0.2. 2.1. Types of maximal singularitiesSet $F=F(f)$, where $f\in\mathcal{F}$, to be a hypersurface satisfying the assumptions of Theorem 0.2. Assume that for some linear system $\Sigma\subset|nH|$ with no fixed components, where $H=-K_F$ is the class of a hyperplane section and $n\geqslant 1$, the pair $(F,\frac{1}{n}D)$, where $D\in\Sigma$ is a general divisor, is not canonical, that is, for some exceptional divisor $E$ over $F$ (depending on $\Sigma$ only, but not on the divisor $D$) the Noether–Fano inequality holds: where $a(E)=a(E,F)$ is the discrepancy of $E$ with respect to $F$. Let us show that this assumption leads to a contradiction. This will prove Theorem 0.2.The exceptional divisor $E$ over $F$ is traditionally called a maximal singularity of the mobile linear system $\Sigma$. Let $B\subset F$ be the centre of the singularity $E$. This is an irreducible subvariety of codimension $\operatorname{codim}(B\subset F)\geqslant 2$. Depending on the value of this codimension and the type of a general point $o\in B$, we have the following options:
We have to eliminate each of these four options. Proposition 2.1. The option (1) does not take place: the inequality $\operatorname{codim}B\geqslant 4$ holds. Proof. Assume that (1) takes place. Then for $M\geqslant 6$ or for $M=5$ and $\operatorname{codim}B=2$, take any irreducible curve $C\subset B$ such that $C\cap\operatorname{Sing}F=\varnothing$. It is well known (see, for instance, [2], Ch. 2, Lemma 2.1 and Proposition 2.3) that for every divisor $D\sim nH$ the inequality $\operatorname{mult}_CD\leqslant n$ holds. Since $B$ is the centre of a maximal singularity of the system $\Sigma$ and $B\not\subset\operatorname{Sing}F$, we obtain $\operatorname{mult}_BD>n$, so that $\operatorname{mult}_CD>n$ too for a general divisor $D\in\Sigma$. We have obtained a contradiction, which proves our claim for $M\geqslant 6$ or $M=5$ and $\operatorname{codim}B=2$.
Let us consider the remaining case when $M=5$ and $\operatorname{codim}B=3$, that is, $B\subset F$ is an irreducible curve on the four-dimensional quintic $F\subset{\mathbb P}^5$. If $B$ does not contain singular points of the hypersurface $F$, then one can argue as above. However, if $B\cap\operatorname{Sing}F\neq\varnothing$, then these arguments do not work. Let $Z=(D_1\circ D_2)$ be the self-intersection of the system $\Sigma$, that is, the algebraic cycle of the scheme-theoretic intersection of the general divisors $D_1,D_2\in\Sigma$. Since $B\subset F$ is a curve, the inequality $\operatorname{mult}_BZ>4n^2$ holds. Furthermore, the degree of the $2$-cycle $Z$ in ${\mathbb P}^5$ is $5n^2$. Therefore, there is a surface $S$ on $F$, satisfying the inequality Consider the case when $B$ is a line in ${\mathbb P}^5$ first. Let $p\in B$ be an arbitrary point which is nonsingular on $F$. If then the intersection $(S\circ T_pF)$ is an effective $1$-cycle in ${\mathbb P}^5$ of degree $5n^2$ satisfying the inequality $\operatorname{mult}_p(S\circ T_pF)>8n^2$, which is impossible. Therefore,However, for points $s,p\in B$ of general position the tangent hyperplanes $T_sF$ and $T_pF$ are distinct, so that $T_sF\cap T_pF\cap F$ is the section of $F$ by some linear subspace of dimension $3$, and by assumption a general point $q\in B$ is nonsingular on this section. Therefore, $S$ is an irreducible component of the $2$-cycle $(T_sF\circ T_pF\circ F)$ of degree $5$ in ${\mathbb P}^5$, which is nonsingular at a general point of the line $B$ but satisfies inequality (4). The only possibility is that the irreducible surface $S$ is a plane. But $F$ does not contain planes. Therefore, the curve $B$ cannot be a line. Let $p,q\in B$ be distinct points which are nonsingular on $F$. Since
$$
\begin{equation*}
\operatorname{mult}_pS+\operatorname{mult}_qS>\frac85\operatorname{deg}S,
\end{equation*}
\notag
$$
the line $[p,q]\subset{\mathbb P}^5$ joining $p$ and $q$ is contained in $S$ and therefore in $F$. If $B$ is a curve in some $2$-plane, then $S$ is this plane $\langle B\rangle$, which lies in $F$, in contradiction to the assumption. Therefore, the linear span $\langle B\rangle$ is not a plane, but then the secant set
$$
\begin{equation*}
\operatorname{Sec}(B)=\overline{\bigcup_{p,q\in B}[p,q]}
\end{equation*}
\notag
$$
is three-dimensional, which is again impossible, as $\operatorname{Sec}(B)\subset S$. (The union is taken over all pairs of distinct points on $B$.) The proof of Proposition 2.1 is complete. Therefore, one of the following options takes place: (2), (3) or (4). Let $o\in B$ be a point of general position. Proposition 2.2. Assume that $o\notin\operatorname{Sing}F$ (that is, the option (2) takes place). Then there is an irreducible subvariety $Y\subset F$ of codimension $3$ satisfying the inequality For a proof, see § 2.3. Proposition 2.3. Assume that the option (3) takes place. Then there is an irreducible subvariety $Y\subset F$ of codimension $2$ satisfying inequality (5). For a proof, see § 2.3. Proposition 2.4. Assume that the option (4) takes place. Then there is an irreducible subvariety $Y\subset F$ of codimension $2$ satisfying the inequality For a proof, see § 2.3. We show now that on the variety $F$ there can be no subvariety $Y$ the existence of which is claimed in Propositions 2.2–2.4. 2.2. Hypertangent divisorsFor a fixed point $o\in F$ and $j\in\{2,\dots,M-1\}$ we denote by $\Lambda_j$ the $j$th hypertangent linear system at this point (see [2], Ch. 3). In the coordinate notation of § 0.1, where $o=(0,\dots,0)$, on the intersection of $F$ with the affine chart ${\mathbb A}^M_{z_1,\dots,z_M}$ the linear system $\Lambda_j$ has the form
$$
\begin{equation*}
\biggl|\sum^j_{i=1}s_{j,j-i}(z_*)f_{[1,i]}(z_*)\big|_F=0\biggr|,
\end{equation*}
\notag
$$
where $f_{[1,i]}=q_1+\dots+q_i$ is the left segment of the polynomial $f$ of length $i$ and the homogeneous polynomials $s_{j,j-i}$ in $z_1,\dots,z_M$ run through the spaces $\mathcal{P}_{j-i,M}$ of homogeneous polynomials of degree ${j-i}$ independently of one another. The technique of hypertangent divisors and the way it is applied to the proof of birational rigidity are well known (see [2], Ch. 3); we only note the key points of the arguments under the assumptions of Propositions 2.2–2.4. Elimination of case (2)Assume that this case takes place. Since $\operatorname{mult}_o Y\leqslant\deg Y$ for every irreducible subvariety, inequality (5) implies that $M\geqslant 9$. The condition (R1) implies that for $j\in\{5,6,\dots,M-1\}$ the inequality
$$
\begin{equation*}
\operatorname{codim}_o(\operatorname{Bs}\Lambda_j\subset F)\geqslant j-4
\end{equation*}
\notag
$$
holds, so that we can construct a sequence of irreducible subvarieties (in the notation of Proposition 2.2), where $\operatorname{codim}(Y_i\subset F)=i$ and for $i\in\{3,\dots,M-6\}$ the subvariety $Y_{i+1}$ is an irreducible component of the effective cycle $(Y_i\circ D_{i+5})$, where $D_{i+5}\in\Lambda_{i+5}$ is a general divisor, with the maximal value of the ratio $\operatorname{mult}_o/\deg$ among all components of that cycle. This construction makes sense because $\operatorname{codim}_o\operatorname{Bs}\Lambda_{i+5}\geqslant i+1$, so that $Y_i$ is not contained in the support of the divisor $D_{i+5}$. For the last subvariety $Y_{M-5}$ in this sequence we obtain the inequality
$$
\begin{equation*}
\operatorname{mult}_oY_{M-5}>\biggl(\frac{8}{M}\,\frac98 \dotsb\frac{M}{M-1}\biggr)\operatorname{deg}Y_{M-5}.
\end{equation*}
\notag
$$
The expression in brackets is equal to $1$, which gives a contradiction ruling out case (2). Elimination of case (3)Assume that this case takes place. Here $M\geqslant 9$ again. Our arguments are completely similar to the arguments in case (2) with some obvious modifications, which we point out below. Here for $j\in\{6,\dots,M-1\}$ by (R2) the inequality
$$
\begin{equation*}
\operatorname{codim}_o(\operatorname{Bs}\Lambda_j\subset F)\geqslant j-5
\end{equation*}
\notag
$$
holds, so that we construct (in precisely the same way) a sequence of irreducible subvarieties where $\operatorname{codim}(Y_i\subset F)=i$ and the last subvariety $Y_{M-6}$ satisfies the inequality
$$
\begin{equation*}
\operatorname{mult}_oY_{M-6}>\biggl(\frac{8}{M}\,\frac98 \dotsb\frac{M}{M-1}\biggr)\operatorname{deg}Y_{M-6},
\end{equation*}
\notag
$$
which is impossible; thus, case (3) is ruled out. Eliminating case (4)Assume that this case takes place. Here $M\geqslant 5$ and for $j\in\{2,\dots,M-1\}$, by (R3) the equality
$$
\begin{equation*}
\operatorname{codim}_o(\operatorname{Bs}\Lambda_j\subset F)=j-1
\end{equation*}
\notag
$$
holds, so that we construct a sequence of irreducible subvarieties where the last subvariety (an irreducible curve) $Y_{M-2}$ satisfies the inequality
$$
\begin{equation*}
\operatorname{mult}_oY_{M-2}>\biggl(\frac{4}{M}\,\frac54 \dotsb\frac{M}{M-1}\biggr)\operatorname{deg}Y_{M-2},
\end{equation*}
\notag
$$
which gives the required contradiction ruling out case (4). The proof of the birational rigidity of the hypersurface $F$ is complete. 2.3. Local inequalitiesConsider again the self-intersection $Z=(D_1\circ D_2)$ of the linear system $\Sigma$, where $D_1,D_2\in\Sigma$ is a general pair of divisors. Under the assumptions of Proposition 2.2 let $P\subset F$ be the section of $F$ by a general $\operatorname{codim}(B\subset{\mathbb P}^M)$-dimensional linear subspace in ${\mathbb P}^M$ containing the point $o\in B$. Since $\operatorname{codim}(B\subset{\mathbb P}^M)\geqslant 5$, we have $\dim P\geqslant 4$ and may assume that $o\in P$ is an isolated centre of a noncanonical singularity of the pair $(P,\frac{1}{n}D_P)$, where $D_P\in\Sigma_P$ is a general divisor and $\Sigma_P=\Sigma|_P$. It is well known (see, for instance, [2], Ch. 2, Theorem 4.1, or [3]) that in this case the $8n^2$-inequality holds: for some linear subspace $\Theta(P)\subset E_P$ of codimension $2$, where $E_P\subset P^+$ is the exceptional divisor of the blow-up $P^+\to P$ of the point $o$ on $P$, the estimate
$$
\begin{equation*}
\operatorname{mult}_o Z_P+\operatorname{mult}_{\Theta(P)} Z^+_P>8n^2
\end{equation*}
\notag
$$
holds, where $Z_P$ is the self-intersection of the linear system $\Sigma_P$ and $Z^+_P$ is its strict transform on $P^+$. Moreover, if $\operatorname{mult}_o Z_P\leqslant 8n^2$, then the subspace $\Theta(P)$ is uniquely determined by the pair $(P,\frac{1}{n}D_P)$ (that is, by the system $\Sigma_P$). Turning back to the original variety $F$ we see that the $8n^2$-inequality holds already on $F$: for some linear subspace $\Theta\subset E_F$ of codimension 2, where $E_F\subset F^+$ is the exceptional divisor of the blow-up $F^+\to F$ of the point $o$ on $F$, the estimate
$$
\begin{equation*}
\operatorname{mult}_o Z+\operatorname{mult}_{\Theta} Z^+>8n^2
\end{equation*}
\notag
$$
holds, where $Z^+$ is the strict transform of $Z$ on $F^+$ and, of course, $\Theta\cap P^+=\Theta(P)$. If $R\subset F$ is a general hyperplane section containing the point $o$ and such that $R^+\supset \Theta$, then it is easy to see that
$$
\begin{equation*}
\operatorname{mult}_o Z_R>8n^2=\frac{8}{M} \operatorname{deg} Z_R,
\end{equation*}
\notag
$$
where $Z_R=(Z\circ R)=Z|_R$ is the self-intersection of the mobile linear system $\Sigma|_R$. By the linearity of the last inequality there is an irreducible component $Y$ of the effective cycle $Z_R$ satisfying (5), which completes the proof of Proposition 2.2. Proposition 2.3 follows directly from the generalized $4n^2$-inequality [4] (see also [8], Ch. I, § 2). Now let us prove Proposition 2.4. In [5] the following general fact was shown. Proposition 2.5 ([5], § 2, Proposition 2.4). Let $X$ be a variety with quadratic singularities of rank at least 4, and assume that $\operatorname{codim}(\operatorname{Sing}X\subset X)\geqslant 4$. Assume further that some divisor $E$ over $X$ is a noncanonical singularity of the pair $(X,\frac{1}{n}\Sigma)$ with centre $B\subset\operatorname{Sing}X$, where $\Sigma$ is a mobile linear system. Then the self-intersection $Z$ of the system $\Sigma$ satisfies the inequality This was shown for quadratic singularities of rank $\geqslant 5$ in [1], § 3; however, the proof there contained an inaccuracy, which was corrected in [5], § 2: in applying the technique of counting multiplicities, instead of the numbers of paths $p_{ij}$ in the graph $\Gamma$ of the maximal singularity, we must use the coefficients $r_{ij}$ that take into account the singularities of the exceptional divisors. The relation between the two groups of integral coefficients was described in detail in [5], § 2. If the rank of the quadratic singularity $o\in F$ is at least $4$, then we apply Proposition 2.5 and obtain
$$
\begin{equation*}
\operatorname{mult}_oZ>4n^2=\frac{4}{M}\operatorname{deg} Z,
\end{equation*}
\notag
$$
which immediately implies Proposition 2.4. Therefore, it remains to consider only the case when $B=\{o\}$ is a quadratic singularity of rank $3$. Let be the resolution of the maximal singularity $E$, that is, the sequence of blow-ups of irreducible subvarieties $B_{j-1}\subset F_{j-1}$, where $j=1, \dots, N$, $F_0=F$, $B_0=\{o\}$, for $j=2, \dots, N$ the subvariety $B_{j-1}$ of codimension $\geqslant 2$ is the centre of $E$ on $F_{j-1}$, and the exceptional divisor $E_N\subset F_N$ of the last blow-up is the centre of $E$ on $F_N$. Set $E_j=\varphi^{-1}_{j,j-1} (B_{j-1})$ for $j=1,\dots,N$. In order that the proof of the $4n^2$-inequality given in [1], § 3 (in the present case, it has the form $\operatorname{mult}_oZ>4n^2$) work for a maximal singularity the centre of which is a point $o$ of rank $3$, the following properties must be satisfied:
For $j=1$ these conditions are satisfied by our assumptions about the hypersurface $F$ (obviously, $a(E_1,F)=M-3\geqslant 2$). Furthermore, as shown in § 1.3, in a neighbourhood of the exceptional divisor $E_1$ the variety $F_1$ has at most quadratic singularities of rank $\geqslant 4$ and $\operatorname{codim} (\operatorname{Sing} F_1\subset F_1)\geqslant 4$. Therefore, the two conditions stated above are satisfied for $j\in\{2, \dots, N\}$ too. This completes the proof of Proposition 2.4. § 3. Codimension of the complementIn this section we prove Theorem 0.3. We consider the violation of each of the conditions defining the subset $\mathcal{F}\subset \mathcal{P}$ and estimate the codimension corresponding to this violation. 3.1. The complement $\mathcal{P}\setminus \mathcal{F}$The set $\mathcal{P}\setminus \mathcal{F}$ consists of the polynomials $f$ such that the hypersurface $F(f)$ does not satisfy one of the conditions of general position listed in § 0.1. In order to prove Theorem 0.3 we have to check that the violation of any of these conditions imposes at least $\gamma(M)$ independent conditions on $f$. First of all, Theorem 3.1 in [5] shows us that the set of polynomials $f\in\mathcal{P}$ such that $\operatorname{Sing} F(f)$ is positive-dimensional is of codimension at least $M(M-2)>\gamma(M)$ in $\mathcal{P}$. Therefore, we may assume that the set $\operatorname{Sing} F(f)$ is finite for all polynomials $f$ under consideration. (We did not include the assumption that the $\operatorname{Sing} F(f)$ is zero-dimensional in the list of conditions of general position in § 0.1 in order to state Theorems 0.1 and 0.2 in the maximal generality.) Furthermore, it is easy to check that $\binom{M-1}{2}+1$ is precisely the codimension of the set of polynomials $f\in\mathcal{P}$ such that some point $o\in F(f)$ is either a quadratic singularity of rank $\leqslant 2$ of the hypersurface $F(f)$, or a singular point of multiplicity ${\geqslant 3}$. Therefore, we can consider only polynomials $f\in\mathcal{P}$ such that the set $\operatorname{Sing} F(f)$ is finite and consists of quadratic singularities of rank $\geqslant 3$. We denote the set of such polynomials by $\mathcal{P}^*$. As we have seen above, $\operatorname{codim}((\mathcal{P}\setminus \mathcal{P}^*)\subset \mathcal{P})\geqslant \gamma(M)$. For a point $o\in{\mathbb P}^M$ let $\mathcal{P}^*(o)\subset \mathcal{P}^*$ be the subset (of codimension $1$) consisting of $f\in \mathcal{P}^*$ such that $f(o)=0$. Denote by $\mathcal{B}_G$, $\mathcal{B}_1$, $\mathcal{B}_2$ and $\mathcal{B}_3$ the subsets of $\mathcal{P}^*(o)$ consisting of $f\in \mathcal{P}^*(o)$ such that, respectively:
In order to prove Theorem 0.3, it is sufficient to check (taking into account that the point $o$ varies in ${\mathbb P}^M$) that the codimension of each of these four subsets of $\mathcal{P}^*(o)$ is at least $\gamma(M)+M-1$. (It is easy to see that the set of four-dimensional quintics that either contain a two-dimensional plane or have a section by a three-dimensional subspace in ${\mathbb P}^5$ with a whole line of singular points, is of codimension higher than $\gamma(5)=6$.) 3.2. Quadratic points of rank 3Let us estimate the codimension of the set $\mathcal{B}_G$ in $\mathcal{P}^*(o)$. For any $M\geqslant 5$ the condition that the point $o$ is a singularity of the hypersurface $F(f)$ means in the notation of § 0.1 that $q_1\equiv 0$ ($M$ independent conditions), and the condition that the rank of the quadratic form $q_2$ is $3$ gives us $\binom{M-2}{2}$ additional independent conditions. We apply the estimate of Theorem 3.1 in [5] again, this time to the cubic hypersurface $\{q_3(0,0,0,z_4,\dots,z_M)=0\}$ in ${\mathbb P}^{M-4}$ for $M\geqslant 8$: we obtain $3(M-6)$ additional independent conditions if the singular locus of this hypersurface is positive-dimensional. Finally, if this singular set is zero-dimensional, then the violation of the last part of condition (G) produces an extra codimension of $1$. Therefore, in the case of the set $\mathcal{B}_G$ for $M\geqslant 8$ it is sufficient to check the inequality which is very easy. For $M\in\{5,6,7\}$ the inequality
$$
\begin{equation*}
\operatorname{codim}(\mathcal B_G\subset \mathcal P^*(o))\geqslant \gamma(M)+M-1
\end{equation*}
\notag
$$
is easy to check too. 3.3. The regularity condition at a nonsingular pointAssume that $M\geqslant 6$. The set $\mathcal{B}_1$ consists of the polynomials $f$ such that the sequence (1) is not regular. We apply the well-known method of estimating the codimension of that set (see [6] or [2], Ch. 3, or [18]): for each $a\in\{6,\dots,M\}$ consider the subset $\mathcal{B}_{1,a}$, consisting of polynomials $f$, such that the regularity of the sequence (1) is violated for the first time by the polynomial $q_a|_{T_oF}$, that is, the set of common zeros of the system of polynomials $q_i$, where $i\in\{6,\dots,a-1\}$, has the correct codimension, whereas $q_a$ vanishes on one of the components of that set. Since the polynomials $q_i$ are homogeneous, it is natural to consider the projectivization ${\mathbb P}(T_oF)\cong {\mathbb P}^{M-2}$. Now we have
$$
\begin{equation*}
\mathcal B_1=\mathcal B_{1,6}\sqcup\dots\sqcup \mathcal B_{1,M}
\end{equation*}
\notag
$$
and the ‘projection method’ (see any of the references above) estimates the codimension of the subsets $\mathcal{B}_{1,a}$ from below by the integers
$$
\begin{equation*}
\binom{M+4}{6},\dots,\binom{M+4}{a},\dots,\binom{M+4}{M}=\binom{M+4}{4},
\end{equation*}
\notag
$$
respectively. It is easy to see that the minimum of this set of integers is $\binom{M+4}{4}$. Therefore, the codimension of $\mathcal{B}_1$ is not less than this number, which is much higher than what we need. 3.4. Quadratic points of higher rankHere $\operatorname{rk} q_2\geqslant 7$ by assumption, so that $q_2\not\equiv 0$; however, $q_1\equiv 0$, which gives $M$ independent conditions. We estimate the codimension of the set $\mathcal{B}_2$, consisting of the polynomials $f$ such that the sequence (2) is not regular, in the same way as in § 3.3: $\mathcal{B}_2=\mathcal{B}_{2,7}\sqcup\cdots\sqcup\mathcal{B}_{2,M}$ and the codimension of the subset $\mathcal{B}_{2,a}$ is at least $\binom{M+5}{a}$, where $a\in\{7,\dots,M\}$. The minimum of these integers is $\binom{M+5}{5}$: this gives us an estimate from below for the codimension of the set $\mathcal{B}_2$, which is much higher than what we need:
$$
\begin{equation*}
\operatorname{codim} (\mathcal B_2\subset \mathcal P^*(o))\geqslant \binom{M+5}{5}+M.
\end{equation*}
\notag
$$
3.5. Quadratic points of lower rankHere $\operatorname{rk} q_2\leqslant 6$, which gives codimension $\binom{M-5}{2}$ (for $M\in\{5,6\}$ the value of this expression is set to be zero); in addition, $q_1\equiv 0$, which results in an extra codimension of $M$. At the same time $\operatorname{rk} q_2\geqslant 3$, so that $q_2\not\equiv 0$ and the sequence (3) is regular in the first term. As in §§ 3.3 and 3.4, we present the set $\mathcal{B}_{3}$ as a union of disjoint subsets:
$$
\begin{equation*}
\mathcal B_{3}=\mathcal B_{3,3}\sqcup \mathcal B_{3,4}\sqcup\dots \sqcup\mathcal B_{3,M},
\end{equation*}
\notag
$$
where $\mathcal{B}_{3,a}$ consists of the polynomials $f$ such that the sequence (3) fails to be regular for the first time in $q_a$. Using the same ‘projection method’ we obtain the estimate
$$
\begin{equation*}
\operatorname{codim} \mathcal B_{3,a}\geqslant \binom{M-5}{2} + \binom{M+1}{a} + M,
\end{equation*}
\notag
$$
$a\in\{3,4,\dots,M\}$. For $a\leqslant M-1$ this estimate is strong enough for our purposes. However, for $a=M$ this is not the case, and we have to estimate the codimension of the set $\mathcal{B}_{3,M}$ separately. We regard the $q_i$, $2\leqslant i\leqslant M$, as homogeneous polynomials on ${\mathbb P}^{M-1}$. By the definition of the set $\mathcal{B}_{3,M}$, if $f\in \mathcal{B}_{3,M}$, then the set of common zeros of the polynomials $q_i$ in ${\mathbb P}^{M-1}$, $2\leqslant i\leqslant M-1$, is zero-dimensional and for some irreducible curve $C\subset{\mathbb P}^{M-1}$, which is a component of this set we have $q_M|_C\equiv 0$. Setting $\dim\langle C\rangle=b\in\{1,\dots,M-1\}$ we obtain the representation
$$
\begin{equation*}
\mathcal B_{3,M}=\mathcal B_{3,M,1}\cup\mathcal B_{3,M,2}\cup \dots \cup\mathcal B_{3,M,M-1},
\end{equation*}
\notag
$$
where $\mathcal{B}_{3,M,b}$ consists of the $f\in\mathcal{B}_{3,M}$ such that $\dim\langle C\rangle=b$. (Generally speaking, this union is not disjoint.) It is sufficient to estimate the codimension of each set $\mathcal{B}_{3,M,b}$ from below. Elementary computations show that
$$
\begin{equation*}
\operatorname{codim} \mathcal B_{3,M,1}\geqslant \binom{M-5}{2}+\binom{M}{2}+M+2.
\end{equation*}
\notag
$$
This is much better than we need. If $b=2$, that is, $C\subset\langle C\rangle\cong{\mathbb P}^2$ is a plane curve of degree $d_C\geqslant 2$, then elementary computations show that the conditions $q_i|_C\equiv 0$ for $i\in\{2,\dots,M\}$ impose conditions on the tuple $\{q_i\}$, that is, on the polynomial $f$ for $d_C=2$ (and an arbitrary plane $\langle C\rangle$), and impose
$$
\begin{equation*}
\sum^{d_C-1}_{i=2}\binom{i+2}{2}+ \sum^M_{i=d_C+1}\biggl( \binom{i+2}{2}-\binom{i+2-d_C}{2}\biggr)
\end{equation*}
\notag
$$
independent conditions for $d_C\in\{3,\dots,M-1\}$, for a fixed plane $\langle C\rangle\subset{\mathbb P}^{M-1}$, which, taking into account the variation of this plane, gives a stronger estimate for the codimension than for $d_C=2$, so that in the end we obtain the inequality
$$
\begin{equation*}
\operatorname{codim} \mathcal B_{3,M,2}\geqslant \binom{M-5}{2}+M^2+1,
\end{equation*}
\notag
$$
which is again much better than what we need. Assume that $b\geqslant 3$. Fix a subspace $P\subset{\mathbb P}^{M-1}$ of dimension $b$ and consider the subset $\mathcal{B}_{3,M,b}(P)$ given by the condition $\langle C\rangle=P$. Here we need the technique of good sequences and associated subvarieties introduced in [7] (see also [2], Ch. 3, § 3, or [5], Definition 3.1). Let us recall briefly this approach. Although the curve $C$ is an irreducible component of the set of common zeros of the polynomials $q_i$, $i\in\{2,\dots,M-1\}$, for $b\leqslant M-2$ it is generally speaking not true that from these polynomials we can choose $b-1$ ones such that $C$ is a component of the set of zeros of their restrictions to $P$. However, it is true that one can choose $b-1$ polynomials $q_{i_1},\dots,q_{i_{b-1}}$, where $i_{\alpha}\in\{2,\dots,M-1\}$, such that there is a sequence of irreducible subvarieties $\Delta_0=P$, $\Delta_1,\dots,\Delta_{b-1}$ of the subspace $P$ such that $\operatorname{codim}(\Delta_{\alpha}\subset P)=\alpha$, $\Delta_{\alpha+1}\subset\Delta_{\alpha}$, $q_{i_{\alpha+1}}|_{\Delta_{\alpha}}\not\equiv 0$ for $\alpha=0,\dots,b-2$, $\Delta_{\alpha+1}$ is an irreducible component of the closed set $\{q_{i_{\alpha+1}}|_{\Delta_{\alpha}}=0\}$ and, moreover, $\Delta_{b-1}=C$. For a fixed tuple $q_{i_1},\dots,q_{i_{b-1}}$ consider the set of all sequences $\Delta_0,\dots,\Delta_{b-1}$ as described above, but with a free end (that is, we drop the last condition $\Delta_{b-1}=C$). However, this set is finite, so that the set of curves coinciding with the last subvariety $\Delta_{b-1}$ in one of such sequences is finite too (and $C$ is one of these curves). Thus, fixing the polynomials $q_{i_1},\dots,q_{i_{b-1}}$ we may assume that the curve $C$ is also fixed. For the condition $q_j|_C\equiv 0$ imposes at least $jb+1$ independent conditions on $q_j$ (since $\langle C\rangle=P$, no polynomial of degree $j$ that is a product of $j$ linear forms on $P$ can vanish identically on $C$; see the details in the above references). The worst estimate for the codimension is obtained when $\{i_1,\dots,i_{b-1}\}=\{M-b+1,\dots,M-1\}$. Therefore, for $b=M-1$ we have $M(M-1)+1$ independent conditions (since $q_M|_C\equiv 0$), and for $3\leqslant b\leqslant M-2$ we have
$$
\begin{equation*}
b(M+2+\dots+M-b)+(M-b)=b\biggl(M+\frac12(M-b+2)(M-b-1)\biggr)+(M-b)
\end{equation*}
\notag
$$
independent conditions. Taking the variation of the subspace $P$ in ${\mathbb P}^{M-1}$ into account we obtain an estimate for the codimension in the form of the integer $h(b)$, where $b\in\{3,\dots,M-1\}$ and Considering the derivative it is easy to check that for $M\in\{5,6\}$ the polynomial $h(t)$ is increasing on the interval $[3,M-1]$, so that the worst estimate for the codimension is $h(3)$. It is easy to see that this estimate is strong enough for the proof of Theorem 0.3 for $M=5,6$. If $M\geqslant 7$, then the behaviour of the function $h(t)$ on the interval $[3,M-1]$ is more complicated: both roots $t_*<t^*$ of the derivative $h'(t)$ lie on this interval, so that $h(t)$ increases on $[3,t_*]$, decreases on the interval $[t_*,t^*]$ and then increases on $[t^*,\infty)$. It is easy to check that $M-2\leqslant t^*\leqslant M-1$, so that the worst estimate from below for the codimension is the minimum of the following three integers:
$$
\begin{equation*}
h(3)=\frac32 M(M-5)+19, \quad h(M-2)=M(M-1)-1 \ \ \text{and}\ \ h(M-1)=M(M-1)+1.
\end{equation*}
\notag
$$
For $M=7$ this minimum is $h(M-2)$, and for $M\geqslant 8$ it is $h(3)$. It is easy to see that this estimate is strong enough. For instance, for $M\geqslant 8$ the codimension is at least which is much higher than what we need. The proof of Theorem 0.3 is complete. Note that for $M\geqslant 8$ the estimate for the codimension of the complement $\mathcal{P}\setminus \mathcal{F}$ in Theorem 0.3 is related only to the violation of the condition for the rank of the quadratic singularity; the violation of condition (G) or the regularity conditions (R1)–(R3) gives a higher codimension.
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