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Forms of del Pezzo surfaces of degree A. V. Zaitsev National Research University Higher School of Economics, Moscow, Russia
Russian version:
Matematicheskii Sbornik, 2023, Volume 214, Number 6, DOI: https://doi.org/10.4213/sm9686 
Bibliographic databases:
    § 1. IntroductionA del Pezzo surface is a smooth projective surface $X$ with ample anticanonical class. The automorphism groups of del Pezzo surfaces over algebraically closed fields of characteristic zero were completely described by Dolgachev and Iskovskikh [1], when they studied finite subgroups of the birational automorphism group of the projective plane. There are only partial results about automorphisms of del Pezzo surfaces over an arbitrary field; the reader can find these in the papers [2] by Dolgachev and Iskovskikh and [3] by Yasinsky (also see [4]). The ideal result would be a complete description of the automorphism groups of del Pezzo surfaces over a prescribed field, analogous to the one obtained in dimension $1$ in [5] and [6]. On the other hand, even the description of groups acting minimally on del Pezzo surfaces will be useful because of its possible applications to the classification of finite subgroups of the group of birational automorphisms of the projective plane over arbitrary fields. The self-intersection index of the canonical class $K_X$ is called the degree of a del Pezzo surface. This index can take integer values from $1$ to $9$. In this paper we consider del Pezzo surfaces of degrees $5$ and $6$ over various fields. In the case of an algebraically (or separably) closed field, a del Pezzo surface of degree $d$ is either isomorphic to $\mathbb{P}^1\times\mathbb{P}^1$, or is obtained by blowing up the projective plane at $9 - d$ points in general position. In particular, for $d = 5, 6$ these surfaces are unique up to isomorphism. In the case of an arbitrary field $K$ an additional invariant appears, namely, an action of the Galois group $\operatorname{Gal} (K^{\mathrm{sep}}/K)$ on the graph of $(-1)$-curves. It is this invariant that we study in our paper. Consider an arbitrary field $K$. Let $X$ be a del Pezzo surface of degree $5$ over $K$. The Galois group $\operatorname{Gal} (K^{\mathrm{sep}}/K)$ acts on the graph of $(-1)$-curves by automorphisms. The automorphism group of this graph is isomorphic to the symmetric group $\mathfrak{S}_5$ (see [7], § 8.5.4), so that we obtain a homomorphism
$$
\begin{equation*}
h\colon\operatorname{Gal} (K^{\mathrm{sep}}/K)\to\mathfrak{S}_5.
\end{equation*}
\notag
$$
The same homomorphism corresponds to the action of the group $\operatorname{Gal}(K^{\mathrm{sep}}/K)$ on the five conic bundle structures on $X$. Given a subgroup $H \subset \mathfrak{S}_5$, we denote its conjugacy class by $[H]$. It is clear that the isomorphism class of a subgroup does not determine its conjugacy class, for example, the subgroup $\langle(12)\rangle$ is not conjugate to $\langle(12)(34)\rangle$ (for clarity, all conjugacy classes are written out at the beginning of § 2). Definition 1.1. We say that a del Pezzo surface of degree $5$ has type $[H]$ if ${\operatorname{Im}{h}\in[H]}$. Note that the isomorphism class of a del Pezzo surface of degree $5$ is not determined by its type in general; see Remark 3.3. A classification of del Pezzo surfaces of degree $5$ up to isomorphisms over fields of characteristic zero was described in [8], Theorem 3.1.3; also see [9], Proposition 4.7, (iv). In this paper we prove the following theorem. Theorem 1.2. There exists a del Pezzo surface of degree $5$ of type $[H]$ over $K$ if and only if there exists a Galois extension of fields $L\supset K$ with Galois group isomorphic to $H$. For some important types of fields Theorem 1.2 gives a simple necessary and sufficient condition for the existence of a del Pezzo surface of degree $5$ of a given type. Corollary 1.3. Let $\mathbb{F}$ be a number field. Then there exist del Pezzo surfaces of degree $5$ of all types over $\mathbb{F}$. Corollary 1.4. Let $\mathbb{F}$ be a finite field. Then there exists a del Pezzo surface of degree $5$ of type $[H]$ over $\mathbb{F}$ if and only if $H$ is a cyclic group. Note that for fields of characteristic zero Theorem 1.2 can be derived from a more accurate result (see [8], Theorem 3.1.3). We prove it in a different way, using a more elementary and more geometric approach which works for arbitrary fields, in particular, imperfect ones. In addition, the advantage of a rougher classification, which is given by Theorem 1.2, is that it is convenient to describe the automorphism groups of del Pezzo surfaces of degree $5$ in such terms. The second main result of this paper is a classification of the automorphism groups of del Pezzo surfaces of degree $5$ depending on their type. Recall that over an algebraically closed field, the automorphism group of the del Pezzo surface of degree $5$ is isomorphic to the group $\mathfrak{S}_5$. Theorem 1.5. The automorphism group of a del Pezzo surface of degree $5$ of type $[H]$ is isomorphic to the centralizer of the subgroup $H$ in the group $\mathfrak{S}_5$. An explicit list of the automorphism groups of del Pezzo surfaces of degree $5$ can be found in § 8. Also note that Theorem 1.5 allows us to obtain, for each subgroup $G\subset\mathfrak{S}_5$, a criterion for the existence of a $G$-minimal del Pezzo surface of degree $5$ over a given field (recall that a del Pezzo surface $X$ with an action of a group $G$ is called $G$-minimal if $\operatorname{rk}\operatorname{Pic}(X)^G=1$). Namely, the following proposition is proved in the paper. Proposition 1.6. Let $G$ be a subgroup of $\mathfrak{S}_5$. Then a $G$-minimal del Pezzo surface of degree $5$ exists if and only if one of the following two conditions is satisfied. 1. $G$ contains a subgroup isomorphic to $\mathbb{Z}/5\mathbb{Z}$. 2. $G$ is trivial and there exists a Galois extension of fields $L\supset K$ such that the group $\operatorname{Gal}(L/K)$ is isomorphic to a subgroup of $\mathfrak{S}_5$ and contains a subgroup isomorphic to $\mathbb{Z}/5\mathbb{Z}$. By analogy with del Pezzo surfaces of degree $5$, one can define the type of a del Pezzo surface of degree $6$ over a field $K$: this is the conjugacy class of the image of the group $\operatorname{Gal}(K^{\mathrm{sep}}/K)$ in the group $\mathfrak{S}_3\times\mathbb{Z}/2\mathbb{Z}$ (see § 9 for more details). As a consequence of Theorem 1.2, we prove the following theorem. Theorem 1.7. There exists a del Pezzo surface of degree $6$ of type $[H]$ over $K$ if and only if there exists a Galois extension of fields $L\supset K$ with Galois group isomorphic to $H$. The plan of the paper is as follows. In § 2 we collect some general statements. In §§ 3 and 4 we consider some ways to construct del Pezzo surfaces of degree $5$. In § 5 we formulate and prove the key lemmas for the proof of Theorem 1.2. In §§ 6 and 7 we prove Theorem 1.2 for infinite and finite fields, respectively. Then, in § 8 we prove Theorem 1.5 and Proposition 1.6. Finally, in § 9, we prove Theorem 1.7. We use the following notation. Let $X$ be an algebraic variety over a field $K$ and $L\supset K$ be a field extension. Then we denote by $X_L$ the extension of the scalars of $X$ to $L$. We denote the algebraic closure of the field $K$ by $\overline{K}$ and its separable closure by $K^{\mathrm{sep}}$. § 2. PreliminariesIn this section we collect some (well-known) general statements that will be convenient to refer to in what follows. But first we list all conjugacy classes of subgroups in the group $\mathfrak{S}_5$ and fix the notation:
Remark 2.1. The subgroups $\mathfrak{S}_3 \times \mathbb{Z}/2\mathbb{Z}$ and $\mathfrak{S}_4$ are maximal (with respect to set inclusion) among the subgroups not containing an element of order $5$. Lemma 2.2. Let $X$ be a del Pezzo surface over a field $K$. Then any $(-1)$-curve on $X_{\overline{K}}$ is defined over $K^{\mathrm{sep}}$. Proof. Let $\mathcal{H}$ be the Hilbert scheme of $(-1)$-curves on $X$. Let us prove that this scheme is smooth. To do this, we calculate the tangent space at the point $[l] \in \mathcal{H}$ corresponding to $(-1)$-curve $l\simeq\mathbb{P}^1$. The tangent space $T_{[l]}\mathcal{H}$ is isomorphic to $H^0(l,\mathcal{N}_{l/X})$ (see [10], Ch. VI, § 4, Theorem 4). Thus, we obtain
$$
\begin{equation*}
T_{[l]}\mathcal{H}\simeq H^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(-1))=0.
\end{equation*}
\notag
$$
Hence the scheme $\mathcal{H}$ is smooth. Therefore, the $K^{\mathrm{sep}}$-points are dense in $\mathcal{H}_{\overline{K}}$ (see [11], Lemma 056U). But $\mathcal{H}_{\overline{K}}$ is finite, and therefore all points of the scheme $\mathcal{H}_{\overline{K}}$ are defined over $K^{\mathrm{sep}}$. The lemma is proved. Lemma 2.3. Let $K$ be a field. Let $X$ be a del Pezzo surface of degree $5$ of type $[H]$ over $K$. Then there is a Galois extension of fields $L\supset K$ with Galois group isomorphic to $H$, and each $(-1)$-curve on $X_{\overline{K}}$ is defined over $L$. Proof. Consider the separable closure of our field. From Lemma 2.2 we know that every $(-1)$-curve on $X_{\overline{K}}$ is defined over $K^{\mathrm{sep}}$. Hence the group $\operatorname{Gal} (K^{\mathrm{sep}}/K)$ acts on the graph of $(-1)$-curves by automorphisms; from this action we obtain a homomorphism $h\colon\operatorname{Gal} (K^{\mathrm{sep}}/K)\to\mathfrak{S}_5$. Denote by $G$ the kernel of this homomorphism. Put $L=(K^{\mathrm{sep}})^G$; then $L\supset K$ is a finite Galois extension with Galois group $\operatorname{Gal}(L/K)\simeq H$ (see [11], Theorem 0BML). Since each $(-1)$-curve is invariant with respect to the action of the group $G$, each $(-1)$-curve on $X_{\overline{K}}$ is defined over $L$. Definition 2.4. Let $K$ be a field. A set of different $K$-points of the projective plane $\mathbb{P}^2_K$ is called points in general position if no three of them are collinear. Lemma 2.5. Let $X$ be a del Pezzo surface of degree $d\leqslant 5$ over a field $K$. Then the natural action of the group $\operatorname{Aut}(X)$ on the graph of $(-1)$-curves of the surface $X_{K^{\mathrm{sep}}}$ by automorphisms is faithful. Proof. Let us show that the action is faithful. Denote the kernel of this action by $G$. Then there exists a $G$-equivariant morphism contracting $9 - d \geqslant 4$ disjoint $(-1)$-curves. The image of these $9 - d$ curves is $9 - d$ points in general position. However, we know that any automorphism of the projective plane fixing four points in general position is trivial. So $G$ is trivial and the action is faithful. The lemma is proved. We denote the intersection graph of $(-1)$-curves on a del Pezzo surface of degree $5$ over an algebraically closed field by $\Gamma$. It is a Kneser graph $KG_{5,2}$ (see Figure 1). That is, its vertices correspond to two-element subsets of a five-element set, and two vertices are adjacent if and only if the corresponding two sets are disjoint. Its automorphism group is isomorphic to the symmetric group $\mathfrak{S}_5$. Corollary 2.6 (see [7], Theorem 8.5.8, or [12], Proposition 3.4). The automorphism group of a del Pezzo surface of degree $5$ embeds in the symmetric group $\mathfrak{S}_5$. We need the following lemma in § 9, when we prove Theorem 1.7. Lemma 2.7. Let $\Gamma$ be the graph of $(-1)$-curves on a del Pezzo surface of degree $5$. Let $G\simeq\mathfrak{S}_3\times\mathbb{Z}/2\mathbb{Z}$ be a subgroup of $\mathrm{Aut}(\Gamma)\simeq\mathfrak{S}_5$. Then there exists a vertex $v$ of this graph that is invariant with respect to the action of $G$. Proof. Note that the graph $\Gamma$ has a vertex invariant with respect to the action of the group $G' = \langle(1,2,3), (1,2), (4,5) \rangle$. Indeed, this is the vertex corresponding to the subset $\{4,5\}$ (see Figure 1). The groups $G$ and $G'$ are conjugate in $\mathfrak{S}_5$, that is, there is a permutation $\sigma\in \mathfrak{S}_5$ such that $G' = \sigma^{-1}G\sigma$. Hence the vertex corresponding to the subset $\{\sigma(4),\sigma(5)\}$ is invariant with respect to the action of $G$. The lemma is proved. The following lemma is proved by similar reasoning. Lemma 2.8. Let $\Gamma$ be the graph of $(-1)$-curves on a del Pezzo surface of degree $5$. Let $G$ be a subgroup of $\mathrm{Aut}(\Gamma)\simeq\mathfrak{S}_5$. Suppose there is no $G$-invariant set of vertices of $\Gamma$ such that no two of these vertices are connected by an edge. Then $G$ contains a subgroup isomorphic to $\mathbb{Z}/5\mathbb{Z}$. Proof. Suppose that $G$ does not contain an element of order $5$; we can assume that $G$ is a maximal subgroup with this property. Then, by Remark 2.1, $G\simeq\mathfrak{S}_3\times\mathbb{Z}/2\mathbb{Z}$ or $G\simeq\mathfrak{S}_4$. In the first case there is a $G$-invariant vertex by Lemma 2.7, in the second case there is a $G$-invariant set of four vertices no two of which are connected by an edge. We obtain a contradiction, so $G$ contains a subgroup isomorphic to $\mathbb{Z}/5\mathbb{Z}$. The lemma is proved.
§ 3. Constructing del Pezzo surfaces of degree $5$. The first constructionIn this section we construct del Pezzo surfaces of degree $5$ of some types by blowing up the projective plane in a closed set. Of this closed set it is required that, after the transition to a separable closure, it become a union of four points in general position. Example 3.1. The surface of type $[e]$ can be constructed over any field $K$. To do this we choose four points in general position on the projective plane $\mathbb{P}^2_K$ (such points exist even over a field of two elements). Blowing up the plane at these four points we obtain a del Pezzo surface of degree $5$. Moreover, all $(-1)$-curves on the surface constructed are defined over $K$, so the group $\operatorname{Gal}(K^{\mathrm{sep}}/K)$ acts on the graph of $(-1)$-curves trivially. Therefore, the surface constructed has type $[e]$. Example 3.2. A surface of type $[\langle(1,2)\rangle]$ can no longer be constructed over an arbitrary field. Let $K$ be a field, and let $L\supset K$ be a Galois extension of fields with Galois group isomorphic to $\mathbb{Z}/2\mathbb{Z}$. We denote by $\omega$ the element generating the extension $L\supset K$ and by $\overline{\omega}$ the image of $\omega$ under the action by a nontrivial element of the Galois group $\operatorname{Gal}(L/K)$. Consider four points in the projective plane $\mathbb{P}^2_{K^{\mathrm{sep}}}$: Remark 3.3. There exist nonisomorphic del Pezzo surfaces of degree $5$ of the same type. Indeed, consider a field $K$ that has two distinct quadratic Galois extensions $L\supset K$ and $L'\supset K$. For each extension you can construct a surface of type $[\langle(1,2)\rangle]$ in the way described in Example 3.2. The resulting del Pezzo surfaces of degree $5$ have the same type, but they are not isomorphic. Example 3.4. Let us use the notation of Example 3.2 and consider four points in the projective plane $\mathbb{P}^2_{K^{\mathrm{sep}}}$: This quadruple forms a closed set defined over $K$. Blowing up the projective plane $\mathbb{P}^2_{K}$ in this closed set, we obtain a del Pezzo surface of degree $5$ of type $[\langle(1,2)(3,4)\rangle]$. Example 3.5. Let $K$ be a field and suppose we have a triple of points in general position on the projective plane $\mathbb{P}^2_{K^{\mathrm{sep}}}$ which is invariant with respect to the action of $\operatorname{Gal}(K^{\mathrm{sep}}/K)$. Assume that we also know that the image of the action of the group $\operatorname{Gal}(K^{\mathrm{sep}}/K)$ in the permutation group of these three points is isomorphic to $\mathbb{Z}/3\mathbb{Z}$. Let us add to this triple a $K$-point such that the resulting four points are in general position over $K^{\mathrm{sep}}$ (any $K$-point suits since the original three points are in general position over $K^{\mathrm{sep}}$ and are cyclically permuted, that is, none of the three lines passing through pairs of these three points can contain a $K$-point). The resulting four points form a closed set defined over $K$. Blowing up the projective plane $\mathbb{P}^2_{K}$ in this closed set we obtain a del Pezzo surface of degree $5$ of type $[\mathbb{Z}/3\mathbb{Z}]$. Remark 3.6. We can obtain only del Pezzo surfaces of degree $5$ of type $[H]$, where $H$ is contained in $G\subset \mathfrak{S}_5$, $G\simeq S_4$, using this construction. § 4. Constructing del Pezzo surfaces of degree $5$. The second constructionConsider an arbitrary field $K$. Let $C$ be a smooth rational conic in the projective plane $\mathbb{P}^2_{K}$. Suppose that there is a quintuple of $K^{\mathrm{sep}}$-points $P_1$, $P_2$, $P_3$, $P_4$ and $P_5$ on $C$ which is invariant with respect to the action of the group $\operatorname{Gal} (K^{\mathrm{sep}}/K)$. Since this tuple of points is invariant with respect to the action of $\operatorname{Gal} (K^{\mathrm{sep}}/K)$, their union is a closed set $Z$ defined over $K$. Let us blow up the plane $\mathbb{P}^2_{K}$ in the set $Z$ and blow down the strict transform of the conic $C$. Then we obtain a surface $X$ which is a del Pezzo surface of degree $5$. Below, in § 7, we will see that this construction gives all possible types of del Pezzo surfaces of degree $5$, except the surface of type $[e]$ over fields of two and three elements. From this construction we clearly see an isomorphism between the group $\mathfrak{S}_5$ and the automorphism group of the graph of $(-1)$-curves on $X_{K^{\mathrm{sep}}}$. When we permute the five specified points on the conic, we also permute ten lines connecting these points, and, as a consequence, we permute the $(-1)$-curves while preserving intersections on the blow-up surface, that is, we obtain an automorphism of the graph. The only permutation of points that acts trivially is the identity permutation. Thus, the action under consideration establishes an isomorphism of $\mathfrak{S}_5$ and the automorphism group of the graph of $(-1)$-curves. Remark 4.1. Let $X$ be a del Pezzo surface of degree $5$ over a field $K$. It is well known that $X$ contains a $K$-point (see [12], for example). Suppose there is a $K$-point that does not lie on $(-1)$-curves (such a $K$-point always exists if the field $K$ is infinite, since in this case the $K$-points on $X$ are Zariski dense (cf. the proof of Theorem 4.4 in [12]), but, for example, there are no such points on the del Pezzo surface of degree $5$ of type $[e]$ over a field of two elements). Then the construction described above is reversible. Namely, we can blow up a point that does not lie on $(-1)$-curves and obtain a surface $X'$, which is a del Pezzo surface of degree $4$. On $X'_{K^{\mathrm{sep}}}$ there are five disjoint $(-1)$-curves invariant with respect to the action of the group $\operatorname{Gal}(K^{\mathrm{sep}}/K)$ (these are precisely the $(-1)$-curves intersecting the exceptional curve $l$ of the blow-up $X'\to X$). Hence the contraction of $X'$ to $\mathbb{P}^2_K$ is defined, and the image of $l$ is a smooth rational conic. This reasoning proves that any del Pezzo surface of degree $5$ over an infinite field can be obtained using the construction described above. These considerations can also be used for an alternative proof of the necessary condition in Theorem 1.2 over infinite fields (cf. [13], § 10). § 5. Points of the affine line and the action of a Galois groupAs we have just seen, to construct a del Pezzo surface of degree $5$ of type $[H]$ it is sufficient to present five points on smooth rational conic whose permutations under the action of the Galois group $\operatorname{Gal}(K^{\mathrm{sep}}/K)$ form a subgroup of $R\subset{\mathfrak{S}_5}$, $R\in[H]$. Since a smooth rational conic is isomorphic to a projective line, it is sufficient to present these five points on $\mathbb{P}^1$ and, in fact, even on the affine line $\mathbb{A}^1$. Now we look at tuples of points on the affine line from the standpoint of the action of $\operatorname{Gal} (K^{\mathrm{sep}}/K)$. The idea of the proof of the following lemma was taken from Rouse’s answer in the discussion [14] on the site https://mathoverflow.net. Lemma 5.1. Given a natural number $n$, a group $G$, and a transitive action of $G$ on the set $\{1,2,\dots, n\}$, let $L\supset K$ be a Galois extension of fields with Galois group isomorphic to $G$. Denote by $H$ the stabilizer of the element $1$ in $G$ and consider the field $M = L^H$. Then the following assertions hold. 1. The extension $M\supset K$ has degree $n$. 2. There exists an element $\beta_1 \in M$ such that $M = K(\beta_1)$. 3. Let $\mu(x)$ be the minimal polynomial of $\beta_1$ over $K$. Then the action of the Galois group $\operatorname{Gal}(K^{\mathrm{sep}}/K)$ on the roots of $\mu(x)$ is equivalent to the action of $G$ on the set $\{1,2,\dots, n\}$. 4. One can number the roots $\beta_1,\beta_2,\dots,\beta_n$ of $\mu(x)$ so that $g(\beta_i) = \beta_{g(i)}$ for any $g\in G$ and $i\in \{1,2,\dots, n\}$. Proof. The degree of the extension $M\supset K$ is calculated as follows:
$$
\begin{equation*}
[M:K]=\frac{[L:K]}{[L:M]}=\frac{|\operatorname{Gal} (L/K)|}{|\operatorname{Gal} (L/M)|}=\frac{|G|}{|H|}=n.
\end{equation*}
\notag
$$
Since the extension $L\supset K$ is separable, the extension $M\supset K$ is separable too. Then by the primitive element theorem (see [15], § 46), the extension $M\supset K$ is simple, that is $M=K(\beta_1)$. Thus, assertions 1 and 2 are proved.
Since the extension $L\supset K$ is normal, the roots of the polynomial $\mu(x)$ lie in the field $L$. So the Galois group $\operatorname{Gal} (K^{\mathrm{sep}}/L)$ acts trivially on these roots, and all nontrivial actions come from the group $\operatorname{Gal}(L/K)\simeq G$. The action of $G$ on the set $\{1,2,\dots, n\}$ is equivalent to its action on the left cosets of the subgroup $H$. Let us show that the action of $G$ on the roots of $\mu(x)$ is equivalent to its action on the left cosets of the same subgroup $H$. Denote by $S$ the stabilizer of the root $\beta_1$. Since the action of $G$ on the roots of $\mu(x)$ is transitive (indeed, we can map the root $\beta_1$ to any root of the $\mu(x)$), we have $[G:S] = n$. Since $\beta_1$ lies in the field $L^H$, the subgroup $H$ is contained in $S$; moreover, the index of the subgroup $H$ in $G$ also equals $n$, and therefore $H = S$. The action of the group $G$ on the roots of $\mu(x)$ is equivalent to the action of $G$ on the left cosets of the subgroup $H$, and this proves assertion 3. Let us number the left cosets $H_1 = H,H_2,\dots,H_n$ in such a way that for any $i\in \{1,2,\dots,n\}$ the class $H_i$ maps the element $1$ to $i$; in this numbering, for any ${g \in G}$ and any $i\in \{1,2,\dots, n\}$ the equality $gH_i = H_{g(i)}$ holds. Now we number the roots as follows
$$
\begin{equation*}
\begin{gathered} \, \beta_2=h_2(\beta_1),\qquad h_2\in H_2; \\ \beta_3=h_3(\beta_1),\qquad h_3\in H_3; \\ \dots \\ \beta_n=h_n(\beta_1),\qquad h_n\in H_n. \end{gathered}
\end{equation*}
\notag
$$
It is easy to see that using this notation, for any $g\in G$ and $i\in \{1,2,\dots, n\}$ we have the equality $g(\beta_i) = \beta_{g(i)}$; thus, assertion 4 is proved.
The lemma is proved. Definition 5.2. Let $G$ be a subgroup of the symmetric group $\mathfrak{S}_n$. The maximum number of orbits of the same length under the action of $G$ on the set $\{1,2,\dots, n\}$ is called the complexity $c(G)$ of the group $G$. Lemma 5.3. Let $n$ and $k$ be natural numbers, and let $G$ be a group of complexity $k$ embedded in $\mathfrak{S}_n$. Suppose a field $K$ has at least $k+1$ elements. If there exists a Galois extension $L\supset K$ with Galois group isomorphic to $G$, then there is a set of $n$ points $B = \{\beta_1, \beta_2, \dots, \beta_n\}$ on the affine line over $L$ such that for any $g\in G$ and $i\in \{1,2,\dots, n\}$ the equality $g(\beta_i) = \beta_{g(i)}$ holds. Proof. We proceed by induction on the number of orbits of the action of $G$ on the set $\{1,2,\dots,n\}$.
Base: one orbit. Since there is only one orbit, $G$ is a transitive subgroup, and we already know how to present the required set of points. Indeed, we take the set $B$ consisting of the roots of the polynomial $\mu(x)$ defined in Lemma 5.1. Step: suppose we are able to present the required set in the case of an action with $m$ orbits; let us present a set for an action with $m + 1$ orbits. Suppose the set $\{1,2,\dots,n\}$ falls into $m+1$ orbits under the action of $G$. Let us choose an orbit $\{i_1, i_2, \dots, i_l\}$ of minimum length; here $l$ is its length. The group $G$ acts transitively on the set $\{i_1, i_2, \dots, i_l\}$. Therefore, according to the induction base, there is a set of $l$ points $\{\beta'_{i_1}, \beta'_{i_2}, \dots, \beta'_{i_l}\}$ on the affine line over $L$ such that for all $g\in G$ and $s\in \{1,2,\dots, l\}$ the equality $g(\beta'_{i_s}) = \beta'_{g(i_s)}$ holds. Moreover, we can assume that $\beta'_{i_1}$ is not equal to zero (this is always the case if $l > 1$; if $l = 1$, then we can take $1$ as $\beta'_{i_1}$). Now consider the set $\{1,2,\dots,n\} \setminus\{i_1, i_2, \dots, i_l\}$; we denote it by $\{j_1, j_2,\dots, j_{n - l}\}$. The group $G$ acts on $\{j_1, j_2, \dots, j_{n - l}\}$ with $m$ orbits, so by the induction assumption there is a set of $n - l$ points $\{\beta_{j_1}, \beta_{j_2}, \dots, \beta_{j_{n-l}}\}$ on the affine line over $L$ such that for any $g\in G$ and any $s\in \{1,2,\dots, n-l\}$ we have $g (\beta_{j_s}) = \beta_{g(j_s)}$. Suppose that $\{\beta'_{i_1}, \beta'_{i_2}, \dots, \beta'_{i_l}\} \subset\{\beta_{j_1}, \beta_{j_2}, \dots, \beta_{j_{n-l}}\}$, while we need a set of $n$ different points. Note that for any $a\in K$ the set $\{a\beta'_{i_1}, a\beta'_{i_2}, \dots, a\beta'_{i_l}\}$ also satisfies the necessary condition, namely, for all $g\in G$ and all $s \in \{1,2,\dots, l\}$ the equality $g(a\beta'_{j_s}) = a\beta'_{g(j_s)}$ holds. Thus, it is sufficient to find a nonzero scalar $a\in K$ such that $a\beta'_{i_1}\notin\{\beta_{j_1}, \beta_{j_2}, \dots, \beta_{j_{n-l}}\}$; then
$$
\begin{equation*}
\{\beta_{j_1},\beta_{j_2},\dots,\beta_{j_{n-l}}\}\cap\{a\beta'_{i_1}, a\beta'_{i_2},\dots, a\beta'_{i_l}\}=\varnothing.
\end{equation*}
\notag
$$
In this case we put $B = \{\beta_{j_1}, \beta_{j_2}, \dots, \beta_{j_{n-l}}\}\cup \{a\beta'_{i_1}, a\beta'_{i_2}, \dots, a\beta'_{i_l}\}$ and obtain the required result.
Suppose there is no nonzero scalar $a\in K$ such that $a\beta'_{i_1}\notin\{\beta_{j_1}, \beta_{j_2}, \dots, \beta_{j_{n-l}}\}$. Then the opposite is true: for any nonzero scalar $a\in K$ the element $a\beta'_{i_1}$ is contained in $\{\beta_{j_1}, \beta_{j_2}, \dots, \beta_{j_{n-l}}\}$. By the hypotheses of the lemma we know that there are at least $k$ nonzero scalars in the field $K$. This means that the set $\{\beta_{j_1}, \beta_{j_2}, \dots, \beta_{j_{n-l}}\}$ contains at least $k$ elements proportional to $\beta'_{i_1}$. Each of these elements lies in its own unique orbit of length $l$, which means that among the orbits of the action of $G$ on the set $\{j_1, j_2, \dots, j_{n - l}\}$ there are at least $k$ orbits of length $l$. Therefore, among the orbits of the action of the group $G$ on the whole set $\{1,2,\dots,n\}$ there are at least $k+1$ orbits of length $l$, which contradicts the condition $c(G) = k$. Thus, we have proved that there exists a nonzero scalar $a\in K$ such that
$$
\begin{equation*}
\{\beta_{j_1},\beta_{j_2},\dots,\beta_{j_{n-l}}\}\cap\{a\beta'_{i_1}, a\beta'_{i_2},\dots, a\beta'_{i_l}\}=\varnothing.
\end{equation*}
\notag
$$
Set
$$
\begin{equation*}
\beta_{i_1}=a\beta'_{i_1}, \quad\beta_{i_2}= a\beta'_{i_2}, \quad\dots, \quad\beta_{i_l}=a\beta'_{i_l},
\end{equation*}
\notag
$$
then we put
$$
\begin{equation*}
\{\beta_{j_1},\beta_{j_2},\dots,\beta_{j_{n-l}}\}\cup\{\beta_{i_1},\beta_{i_2},\dots,\beta_{i_l}\}=\{\beta_1,\beta_2,\dots,\beta_n\}
\end{equation*}
\notag
$$
and for any $g\in G$ and any $i\in \{1,2,\dots, n\}$ the equality $g(\beta_i) = \beta_{g(i)}$ holds.
Lemma 5.3 is proved. Corollary 5.4. Let $n$ and $k$ be natural numbers, and let $G$ be a group of complexity $k$ embedded in $\mathfrak{S}_n$. Suppose a field $K$ has at least $k+1$ elements. If there exists a Galois extension $L\supset K$ with Galois group isomorphic to $G$, then there is a set of $n$ points $\{\beta_1, \beta_2, \dots, \beta_n\}$ on the affine line over $L$ such that the action of the Galois group $\operatorname{Gal}(K^{\mathrm{sep}}/K)$ on this set is equivalent to the action of $G$ on the set $\{1,2,\dots,n\}$. Proof. Using Lemma 5.3 we obtain a set of points $\{\beta_1, \beta_2, \dots, \beta_n\}$ on $\mathbb{A}^1_L$ and, in particular, on $\mathbb{A}^1_{K^{\mathrm{sep}}}$. From the condition that for any $g\in G$ and any $i\in \{1,2,\dots, n\}$ we have $g(\beta_i) = \beta_{g(i)}$, it immediately follows that the actions of the group $G$ on the sets $\{1,2,\dots,n\}$ and $\{\beta_1, \beta_2, \dots, \beta_n\}$ are equivalent.
For any $i \in \{1,2,\dots, n\}$ the element $\beta_i$ lies in the field $L$, thus, the action of the group $\operatorname{Gal}(K^{\mathrm{sep}}/K)$ on the set $\{\beta_1, \beta_2, \dots, \beta_n\}$ is induced by the action of the group $\operatorname{Gal}(L/K)\simeq G$ via the homomorphism $\operatorname{Gal}(K^{\mathrm{sep}}/K)\to \operatorname{Gal}(L/K)$, hence it is equivalent to the action of the group $G$ on the set $\{1,2,\dots,n\}$. Corollary 5.5. The statement of Corollary 5.4 remains true if the affine line is replaced by the projective one. In fact, we use Corollary 5.4 and then embed the affine line in the projective line. Corollary 5.6. Let $n$ and $k$ be natural numbers, and let $G$ be a group of complexity $k$ embedded in $\mathfrak{S}_n$. Suppose a field $K$ has at least $k+1$ elements. If there exists a Galois extension $L\supset K$ with Galois group isomorphic to $G$, then there is a smooth conic $C$ in the projective plane $\mathbb{P}^2_{K^{\mathrm{sep}}}$ and a set of $n$ points on $C$, which is invariant with respect to the action of the group $\operatorname{Gal}(K^{\mathrm{sep}}/K)$ and such that the action of $\operatorname{Gal}(K^{\mathrm{sep}}/K)$ on this set is equivalent to the action of $G$ on the set $\{1,2,\dots,n\}$. In fact, we can use Corollary 5.5 and embed the projective line isomorphically in the projective plane as a smooth conic. § 6. Realization of types of del Pezzo surfaces of degree $5$ over infinite fieldsNow we are ready to prove Theorem 1.2 on the realization of del Pezzo surfaces of degree $5$ of various types over infinite fields. Proof of Theorem 1.2. The necessity proof was given in Lemma 2.3. Note that there are no restrictions on the original field in Lemma 2.3, so in fact necessity was also proved for finite fields.
Now we prove the sufficiency of the condition. We are given a subgroup $H\subset\mathfrak{S}_5$, and we know that the field $K$ has a Galois extension $L\supset K$ with Galois group isomorphic to $H$. We want to construct a del Pezzo surface of degree $5$ of type $[H]$. To do this, we use Corollary 5.6 for $n= 5$, $k=c(H)$ and $G=H$ and also the construction described in § 4. The resulting surface is a del Pezzo surface of degree $5$ of type $[H]$. The theorem is proved. Proof of Corollary 1.3. It is sufficient to prove that for any subgroup $H\subset\mathfrak{S}_5$ there exists a Galois extension of fields $\mathbb{F}'\supset\mathbb{F}$ with Galois group isomorphic to $H$, and then use Theorem 1.2 in the case of an infinite field considered above.
Note that all subgroups of the group $\mathfrak{S}_5$ are solvable except $\mathfrak{A}_5$ and $\mathfrak{S}_5$. The existence of a Galois extension of a number field with a prescribed solvable Galois group was proved by Shafarevich (see [16], Theorem 7). The proof of the existence of Galois extensions of a number field with Galois groups isomorphic to $\mathfrak{A}_5$ and $\mathfrak{S}_5$ was presented in [17], § 1. § 7. Realization of types of del Pezzo surfaces of degree $5$ over finite fieldsIn fact, the case of a finite field is not much different from the case of an infinite field. For example, for fields with seven elements or more the proof of Theorem 1.2 literally coincides with the proof in the previous section. In the case of a field with two, three, four or five elements there are some nuances, but the theorem remains true. Proof of Theorem 1.2 (finite field). The proof of the necessary condition of the theorem for finite fields, as already discussed, is exhausted by Lemma 2.3, so we only need to prove sufficiency.
Thus, suppose we have a subgroup $H\subset\mathfrak{S}_5$ and we also know that the field $K$ has a Galois extension $L\supset K$ with Galois group isomorphic to $H$. We want to construct a del Pezzo surface of degree $5$ of type $[H]$. We need to consider the following five cases. 1. $|K|>c(H)$. We use Corollary 5.6 for $n=5$, $k=c(H)$ and $G=H$ and also the construction described in § 4. The resulting surface is a del Pezzo surface of degree $5$ of type $[H]$. 2. $|K|=5=c(H)$. It follows from the equality $c(H)=5$ that the group $H$ is trivial. The construction of a del Pezzo surface of degree $5$ of type $[e]$ was described in Example 3.1. 3. $|K|=4\leqslant c(H)$. It follows from the inequality $c(H)\geqslant 4$ that the group $H$ is trivial. The construction of a degree $5$ del Pezzo surface of type $[e]$ was described in Example 3.1. 4. $|K|=3\leqslant c(H)$. It follows from the inequality $c(H)\geqslant 3$ that $H$ is either the trivial subgroup or a subgroup of order $2$ generated by a transposition. Constructions of del Pezzo surfaces degree $5$ of type $[e]$ and type $[\langle(1,2)\rangle]$ were described in Examples 3.1 and 3.2. 5. $|K|=2\leqslant c(H)$. Since $H$ is the Galois group of a finite extension of a finite field, $H$ is cyclic. The inequality $c(H)\geqslant 2$ holds for the following cyclic subgroups.
All cases have now been considered, and thus Theorem 1.2 is proved in full generality without any restrictions on the base field. § 8. Automorphism groups of del Pezzo surfaces of degree $5$We have dealt with criteria for the existence of del Pezzo surfaces of degree $5$ of various types over a given field, and now it is natural to ask about the automorphism groups of these surfaces. Let $X$ be a del Pezzo surface of degree $5$ of type $[H]$ over a field $K$. We have noticed many times that the group $\operatorname{Gal} (K^{\mathrm{sep}}/K)$ acts on the graph of $(-1)$-curves by automorphisms; we denoted the corresponding homomorphism to the group $\mathfrak{S}_5$ by $h$. By Lemma 2.5 the group $\mathrm{Aut}(X)$ also acts on the graph of $(-1)$-curves by automorphisms. Denote the corresponding homomorphism to $\mathfrak{S}_5$ by $\psi$; we recall from Corollary 2.6 that $\psi$ is an injection. Lemma 8.1. The subgroup $\psi(\mathrm{Aut}(X))$ lies in the centralizer of the subgroup $h(\operatorname{Gal}(K^{\mathrm{sep}}/K))$. Proof. Automorphisms of the surface $X$ commute with the action of the Galois group $\operatorname{Gal}(K^{\mathrm{sep}}/K)$ on this surface. Consequently, the subgroups $\psi(\mathrm{Aut}(X))$ and $h(\operatorname{Gal}(K^{\mathrm{sep}}/K))$ commute, that is, $\psi(\mathrm{Aut}(X))$ lies in the centralizer of the subgroup $h(\operatorname{Gal}(K^{\mathrm{sep}}/K))$. The lemma is proved. Notation 8.2. Let $H$ be a subgroup of a group $G$. Then we denote by $C_G(H)$ the centralizer of $H$ in $G$. Now we are ready to prove Theorem 1.5. Proof of Theorem 1.5. Let $X$ be a del Pezzo surface of degree $5$ of type $[H]$ over a field $K$. Without loss of generality we can assume that $h(\operatorname{Gal}(K^{\mathrm{sep}}/K)) = H$. We know from Lemma 8.1 that $\psi(\mathrm{Aut}(X))\subset C_{\mathfrak{S}_5}(H)$. It remains to show that these groups actually coincide.
Consider the permutation $\sigma\in C_{\mathfrak{S}_5}(H)$. Since each $(-1)$-curve on $X_{\overline{K}}$ is defined over $K^{\mathrm{sep}}$, there exists an automorphism $f$ of the surface $X_{K^{\mathrm{sep}}}$ such that $\psi(f) = \sigma$. Denote by $l_1,l_2,\dots,l_{10}$ all $(-1)$-curves on the surface $X_{K^{\mathrm{sep}}}$. Since $\sigma$ lies in the centralizer of the subgroup $H$, for any $\gamma\in\operatorname{Gal}(K^{\mathrm{sep}}/K)$ and any $i\in \{1,2,\dots, 10\}$ the equality holds. Here $\gamma(f)$ is the automorphism obtained by applying the element $\gamma$ to the automorphism $f$. From the fact that the set $\{\gamma(l_i)\}_{i=1,2,\dots,10}$ coincides with $\{l_i\}_{i=1,2,\dots,10}$ we conclude that the automorphisms $f$ and $\gamma(f)$ induce the same automorphism of the graph of $(-1)$-curves. Then by Lemma 2.5 these automorphisms coincide:Since the above equality holds for any $\gamma \in\operatorname{Gal}(K^{\mathrm{sep}}/K)$, using the equality we conclude that $f$ is defined over $K$. Therefore, each permutation $\sigma\in C_{\mathfrak{S}_5}(H)$ is realized by an automorphism of the surface $X$, that is, $\psi(\mathrm{Aut}(X)) = C_{\mathfrak{S}_5}(H)$. Since $\psi$ is injective, we obtain $\mathrm{Aut}(X)\simeq C_{\mathfrak{S}_5}(H)$, as required. The theorem is proved.Corollary 8.3. Let $X$ be a del Pezzo surface of degree $5$. Then, depending on the type, $X$ has the automorphism groups presented in Table 1. Table 1.
To prove the corollary we use Theorem 1.5 and calculate the centralizers $C_{\mathfrak{S}_5}(H)$ for all types of surfaces. Recall that a del Pezzo surface $X$ endowed with an action of a $G$ is called $G$-minimal if $\operatorname{rk}\operatorname{Pic}(X)^G=1$. Lemma 8.4 (cf. [1], Theorem 6.4). Let $X$ be a del Pezzo surface of degree $5$ over a field $K$, and let $G$ be a subgroup of $\mathrm{Aut}(X)\subset\mathfrak{S}_5$. Then $X$ is $G$-minimal if and only if the group $\Delta$ generated by $G$ and the image of the group $\operatorname{Gal}(K^{\mathrm{sep}}/K)$ under the homomorphism $h$ contains a subgroup isomorphic to $\mathbb{Z}/5\mathbb{Z}$. Proof. Consider the following exact sequence of groups:
$$
\begin{equation*}
0\to \operatorname{Pic}(X)\to \operatorname{Pic}(X_{K^{\mathrm{sep}}})^{\operatorname{Gal}(K^{\mathrm{sep}}/K)}\to \operatorname{Br}(X)\to \operatorname{Br}(K(X))
\end{equation*}
\notag
$$
(see [18], Exercise 3.3.5, (iii)). Now, since there is a $K$-point on $X$ (see, for example, [12]), the last homomorphism is an embedding, hence Going over to $G$-invariants we obtain an isomorphism:
If $X$ is $G$-minimal, then by Lemma 2.8 the subgroup $\Delta$ contains a group isomorphic to $\mathbb{Z}/5\mathbb{Z}$. Conversely, if $\Delta$ contains a group $\mathbb{Z}/5\mathbb{Z}$, then $\operatorname{rk}(\operatorname{Pic}(X_{K^{\mathrm{sep}}})^{\Delta}) = 1$, because a five-dimensional representation of $\mathbb{Z}/5\mathbb{Z}$ over $\mathbb{Q}$ that has a one-dimensional trivial subrepresentation is either trivial itself (which is not our case), or contains a unique one-dimensional trivial subrepresentation. Therefore, $X$ is $G$-minimal. The lemma is proved. From Lemmas 8.1 and 8.4 we obtain a corollary. Corollary 8.5. The automorphism group of a minimal del Pezzo surface of degree $5$ is either trivial or isomorphic to $\mathbb{Z}/5\mathbb{Z}$. Now let us prove Proposition 1.6. Proof of Proposition 1.6. Let us prove necessity. Suppose that there exists a $G$-minimal del Pezzo surface of degree $5$ over the field $K$. From Lemma 8.4 we know that the group $\Delta$ generated by $G$ and the image of $\operatorname{Gal}(K^{\mathrm{sep}}/K)$ under the homomorphism $h$ contains a subgroup isomorphic to $\mathbb{Z}/5\mathbb{Z}$. Since the subgroups $G\subset\mathfrak{S}_5$ and $h(\operatorname{Gal}(K^{\mathrm{sep}}/K))\subset\mathfrak{S}_5$ commute, there is a surjective homomorphism
$$
\begin{equation*}
p\colon G\times h(\operatorname{Gal}(K^{\mathrm{sep}}/K))\twoheadrightarrow\Delta.
\end{equation*}
\notag
$$
Therefore, either $G$ contains a subgroup isomorphic to $\mathbb{Z}/5\mathbb{Z}$, or $h(\operatorname{Gal}(K^{\mathrm{sep}}/K))$ contains a subgroup isomorphic to $\mathbb{Z}/5\mathbb{Z}$. In the first case condition 1 in Proposition 1.6 holds. In the second case condition 2 holds by Theorem 1.2. Moreover, by Theorem 1.5 the subgroup $G$ is contained in the centralizer $C_{\mathfrak{S}_5}(h(\operatorname{Gal}(K^{\mathrm{sep}}/K)))$ which lies in the centralizer $C_{\mathfrak{S}_5}(\mathbb{Z}/5\mathbb{Z}) \simeq \mathbb{Z}/5\mathbb{Z}$. If condition 1 does not hold, that is, $G$ does not contain $\mathbb{Z}/5\mathbb{Z}$, then $G$ is trivial.
Now let us prove sufficiency. Suppose condition 1 holds. Then the del Pezzo surface of degree $5$ of type $[e]$ is $G$-minimal, and by Theorem 1.2 a surface of type $[e]$ exists over any field ($G$-minimality follows from Lemma 8.4). Assume that condition 2 holds. Then by Theorem 1.2 there exists a del Pezzo surface of degree $5$ of type $[\operatorname{Gal}(L/K)]$ over $K$. By Lemma 8.4 the surface $X$ is $G$-minimal with respect to the action of the trivial group $G$. The proof is complete. § 9. Classification of del Pezzo surfaces of degree $6$Consider an arbitrary field $K$. Let $X$ be a del Pezzo surface of degree $6$ over $K$. The Galois group $\operatorname{Gal}(K^{\mathrm{sep}}/K)$ acts on the graph of $(-1)$-curves by automorphisms. Since the automorphism group of the graph of $(-1)$-curves of $X_{K^{\mathrm{sep}}}$ is isomorphic to $\mathfrak{S}_3\times\mathbb{Z}/2\mathbb{Z}$ (see [7], Theorem 8.4.2), we obtain a homomorphism
$$
\begin{equation*}
h\colon\operatorname{Gal} (K^{\mathrm{sep}}/K)\to\mathfrak{S}_3\times\mathbb{Z}/2\mathbb{Z}.
\end{equation*}
\notag
$$
Let $H$ be a subgroup in $\mathfrak{S}_3\times\mathbb{Z}/2\mathbb{Z}$ and denote its conjugacy class by $[H]$. Definition 9.1. We say that a del Pezzo surface of degree $6$ has type $[H]$ if ${\operatorname{Im}{h}\in [H]}$. It is easy to see that there are only ten types of del Pezzo surfaces of degree $6$, namely, $[e]$, $[\langle((1,2),0)\rangle]$, $[\langle((1,2),1)\rangle]$, $[\langle(\mathrm{id},1)\rangle]$, $[\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/2\mathbb{Z}]$, $[\mathbb{Z}/3\mathbb{Z}]$, $[\mathbb{Z}/6\mathbb{Z}]$, $[\langle((123),0), ((12),0)\rangle]$, $[\langle((123),0), ((12),1)\rangle]$, and $[\mathfrak{S}_3\times\mathbb{Z}/2\mathbb{Z}]$ (cf. [19], Figure 1). Proof of Theorem 1.7. The proof of the necessity of the condition in question is quite analogous to the proof of the necessity of the condition in Theorem 1.2. Therefore, we immediately move on to sufficiency.
Suppose we have a subgroup $H\subset\mathfrak{S}_3\times\mathbb{Z}/2\mathbb{Z}$ and we also know that the field $K$ has a Galois extension $L\supset K$ with Galois group isomorphic to $H$. We want to construct a del Pezzo surface of degree $6$ of type $[H]$. To do this, first we construct a del Pezzo surface of degree $5$ of the corresponding type. Consider an embedding
$$
\begin{equation*}
i\colon\mathfrak{S}_3\times\mathbb{Z}/2\mathbb{Z}\hookrightarrow\mathfrak{S}_5.
\end{equation*}
\notag
$$
Denote the image of this homomorphism by $G$, put $H' = i(H)$ and consider a del Pezzo surface of degree $5$ of type $[H']$. The existence of such a surface is guaranteed by Theorem 1.2. Since $H'$ is a subgroup of $G$, by Lemma 2.7 there is a vertex $v$ of the graph $\Gamma$ that is invariant with respect to the action of $H'$. So the $(-1)$-curve corresponding to the vertex $v$ is invariant with respect to the action of $\operatorname{Gal}(K^{\mathrm{sep}}/K)$, and therefore is defined over $K$. Blowing down this $(-1)$-curve we obtain a del Pezzo surface of degree $6$ of type $[H]$. The theorem is proved. A classification of rational del Pezzo surfaces of degree $6$ (in particular, $G$-minimal ones) over perfect fields and some information about their automorphism groups can be found in [19], § 4.
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