| Sbornik: Mathematics |
|
|
|
|
|
Belyi's theorem for smooth complete intersections of general type in generalised Grassmannians and weighted projective spaces M. A. Ovcharenkoab a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia b International Laboratory for Mirror Symmetry and Automorphic Forms, National Research University Higher School of Economics, Moscow, Russia
Bibliographic databases:
    § 1. IntroductionIn this paper we obtain a Belyi-type characterisation of smooth complex complete intersections of general type in generalised Grassmannians and weighted projective spaces which can be defined over $\overline{\mathbb{Q}}$. Throughout the paper all varieties are assumed to be defined over $\mathbb{C}$. A smooth projective variety $X$ is of general type if its Kodaira dimension is maximum possible, that is, equals $\dim(X)$. Recall Belyi’s classical theorem. Theorem 1.1 (see [3] and [17]). A smooth complex projective curve $X$ can be defined over $\overline{\mathbb{Q}}$ if and only if there exists a surjective morphism $f\colon X \to \mathbb{P}^1$ that is étale over $\mathbb{P}^1(\mathbb{C}) \setminus \{0, 1, \infty\}$. Belyi’s theorem is a starting point for Grothendieck’s theory of dessins d’enfants: it implies that $\operatorname{Gal}(\mathbb{Q})$ acts faithfully on the étale fundamental group of ${\mathbb{P}^1(\mathbb{C}) \setminus \{0,1, \infty\}}$ by outer automorphisms (see [32], Theorem 4.7.7). Moreover, $\operatorname{Gal}(\mathbb{Q})$ acts faithfully on the set of connected components of the moduli space of surfaces of general type (see [2] and [9]). Thus it is natural to expect higher-dimensional analogues of Belyi’s theorem. In the case when $\dim(X) \geqslant 2$ a morphism $X \to \mathbb{P}^1$ is replaced by a Lefschetz function. Definition 1.2. Let $X$ be a smooth projective variety, and let $\mathcal{H}$ be a very ample linear system on $X$. A Lefschetz pencil $\mathcal{L} \subset \mathcal{H}$ is a one-dimensional linear subsystem such that $\operatorname{codim}_X(\operatorname{Bs}(\mathcal{L})) > 1$ and an element $X_t \in \mathcal{L}$ of the pencil has at worst a single ordinary double point. A Lefschetz function (see [11] and [15]) is a composition of the rational map $X \dashrightarrow \mathbb{P}^1$ defined by a Lefschetz pencil with a rational function on $\mathbb{P}^1$. Remark 1.3. In Theorem 1.1 we can assume that all ramification indices of $f$ at points lying over 1 are equal to 2 (see [32], Exercise 4.7). Thus, Definition 1.2 is consistent with the one-dimensional case. A Belyi-type theorem holds for smooth surfaces of general type by a result of González-Diez (see [11], Theorem 1 and Proposition 1). Following the approach of González-Diez, Javanpeykar obtained a Belyi-type theorem for smooth complete intersections of general type in ordinary projective spaces. Theorem 1.4 ([15], Theorem 1.1). Let $X \subset \mathbb{P}^n$ be a smooth complex complete intersection of general type of dimension at least 3. Then the variety $X$ can be defined over $\overline{\mathbb{Q}}$ if and only if there exists a Lefschetz function $X \dashrightarrow \mathbb{P}^1$ with at most three critical points. In this paper we show that the proof of Theorem 1.4 can be modified for smooth complete intersections of general type in generalised Grassmannians and weighted projective spaces, using the existing results on the infinitesimal Torelli theorem for such varieties (see [18] and [37]). We refer the reader to § 2 for preliminaries and the notation for (weighted) complete intersections. Definition 1.5. A generalised Grassmannian is a projective rational homogeneous variety with Picard number 1. Remark 1.6. Any generalised Grassmannian can be presented as $G / P$, where $G$ is a simple algebraic group, and $P$ is a maximal parabolic subgroup (see [21], § 2, for a survey of rational homogeneous spaces). Let us formulate the main results of the paper. Theorem 1.7. Let $Y$ be a generalised Grassmannian and $X \subset Y$ be a smooth complex complete intersection of general type of dimension at least 3. Then the variety $X$ can be defined over $\overline{\mathbb{Q}}$ if and only if there exists a Lefschetz function $X \dashrightarrow \mathbb{P}^1$ with at most three critical points. Theorem 1.8. Let $X \subset \mathbb{P}(\rho)$ be a smooth well-formed complex weighted complete intersection of general type of dimension at least 3. Assume that $\dim(|\mathcal{O}_X(1)|) \geqslant 2$. Then the variety $X$ can be defined over $\overline{\mathbb{Q}}$ if and only if there exists a Lefschetz function $X \dashrightarrow \mathbb{P}^1$ with at most three critical points. Remark 1.9. The only reason we assume that $\dim(|\mathcal{O}_X(1)|) \geqslant 2$ is because this is crucial for Usui’s proof of the infinitesimal Torelli theorem (see [37]). We expect that this restriction can be dropped (see Question 3.4). The structure of the paperIn § 2 we briefly recall some basic facts about (weighted) complete intersections and Cox rings. In § 3 we prove Theorems 1.7 and 1.8. In § 4 we discuss their possible generalisation to smooth well-formed complete intersection of general type in weighted generalised Grassmannians. Notation and conventionsFor a variety $Y$ we denote by $\operatorname{WDiv}(Y)$ its group of Weil divisors, and for a field $\Bbbk$ we denote by $Y(\Bbbk)$ the set of $\Bbbk$-points of $Y$. All varieties in our papers are assumed to be defined over $\mathbb{C}$. Note that most arguments also hold for an arbitrary algebraically closed field on characteristic 0. AcknowledgementsThe author is grateful to C. Shramov for introducing us to the paper [15] and for useful discussions, to A. Javanpeykar for encouraging us to write this paper and valuable comments, and to V. Przyjalkowski for many helpful suggestions and for reading the paper carefully. § 2. Preliminaries2.1. Complete intersections and Cox ringsThe goal of this subsection is to recall some generalities on complete intersections and Cox rings. 2.1.1. Cox rings and Mori dream spacesDefinition 2.1. Let $Y$ be an irreducible normal variety with $\operatorname{Cl}(Y) \cong \mathbb{Z}^{\rho}$. Fix a subgroup $K \subset \operatorname{WDiv}(Y)$ such that the canonical map $K \to \operatorname{Cl}(Y)$ taking a Weil divisor $D \in K$ to its class $[D] \in \operatorname{Cl}(Y)$ is an isomorphism. The Cox ring of the variety $Y$ is the following $\operatorname{Cl}(Y)$-graded algebra: where multiplication in $\mathcal{R}(Y)$ is defined by multiplying homogeneous sections in the field of rational functions $\mathbb{C}(Y)$. Remark 2.2. The Cox ring $\mathcal{R}(Y)$ is unique up to isomorphism, and $\mathcal{R}(Y)$ is an integral domain (see [1], Construction 1.4.1.1 and § 1.5.1). Definition 2.3 (see [1], Definition 3.3.4.1). An irreducible normal projective variety $Y$ with divisor class group $\operatorname{Cl}(Y) \cong \mathbb{Z}^{\rho}$ and finitely generated Cox ring $\mathcal{R}(Y)$ is a Mori dream space of rank $\rho$. Remark 2.4. Each Mori dream space has a natural decomposition of its cone of effective divisors into convex polyhedral sets, called Mori chambers. It encodes the birational geometry of the Mori dream space (see [1], § 3.3.4). Example 2.5. Any smooth Fano variety is a Mori dream space (see [1], Theorem 4.3.3.7). The Cox ring of a generalised flag variety $G / P$ can be described in terms of the representation theory of the group $G$ (see [1], § 3.2.3). 2.1.2. The correspondence between closed subschemes and homogeneous idealsThere exists a natural correspondence between closed subschemes of a smooth Mori dream space $Y$ and homogeneous ideals in its Cox ring $\mathcal{R}(Y)$. More precisely, ideal sheaves on $Y$ correspond to so-called ‘saturated’ homogeneous ideals in $\mathcal{R}(Y)$. Notation 2.6 (see [24], Definition 2.2). Let $Y$ be an irreducible normal variety with divisor class group $\operatorname{Cl}(Y) \cong \mathbb{Z}^{\rho}$. For a homogeneous element $f \in \mathcal{R}(Y)$ we denote by $D_f$ the corresponding effective divisor on $Y$. 1. For a homogeneous ideal $I \subset \mathcal{R}(Y)$ we denote by $\varphi_Y(I)$ the ideal sheaf associated with the following sum of $\mathcal{O}_Y$-subsheaves:
$$
\begin{equation*}
\varphi_Y(I)=\sum_{\substack{f \in I \\ \text{homogeneous}}} \mathcal{O}_Y(-D_f) \subseteq \mathcal{O}_Y.
\end{equation*}
\notag
$$
In particular, $\varphi_Y(I)$ corresponds to a unique closed subscheme in $Y$ (for example, see [13], Proposition II.5.9). 2. For an ideal sheaf $\mathcal{I} \subseteq \mathcal{O}_Y$ let $\psi_Y(\mathcal{I}) \subset \mathcal{R}(Y)$ denote the ideal generated by all homogeneous elements $f \in \mathcal{R}(Y)$ such that $\mathcal{O}_Y(-D_f) \subseteq \mathcal{I}$. Remark 2.7. If $\operatorname{Cl}(Y) \simeq \mathbb{Z}$, then $Y \simeq \operatorname{Proj}(\mathcal{R}(Y))$ (see [1], § 3.3.4). For any homogeneous ideal $I \subset \mathcal{R}(Y)$ the quotient map $\mathcal{R}(Y) \twoheadrightarrow \mathcal{R}(Y) / I$ defines the closed subscheme $\operatorname{Proj}(\mathcal{R}(Y) / I) \subset \operatorname{Proj}(\mathcal{R}(Y)) = Y$. Its ideal sheaf can be identified with the ideal sheaf $\varphi_Y(I)$ introduced above (for example, see [13], Proposition II.5.9). Proposition 2.8. Let $Y$ be a smooth Mori dream space, $\mathcal{R}(Y)$ be its Cox ring, and $B \subset \mathcal{R}(Y)$ be the irrelevant ideal. (1) Let $I = (f_1, \dots, f_m) \subset \mathcal{R}(Y)$ be a homogeneous ideal. Then the following decomposition holds: $\varphi_Y(I) = \sum_{j = 1}^m \mathcal{O}_Y(-D_{f_j})$. (2) For any homogeneous ideal $I \subset \mathcal{R}(Y)$ we have $\psi_Y(\varphi_Y(I)) = (I : B^{\infty})$, where $(I : B^{\infty})$ is the saturation of the ideal $I$ with respect to $B$:
$$
\begin{equation*}
(I : B^{\infty})=\bigcup_{l=1}^{\infty} \{f \in \mathcal{R}(Y) \mid f B^{l} \subset I\}.
\end{equation*}
\notag
$$
(3) For an ideal sheaf $\mathcal{I} \subseteq \mathcal{O}_Y$ we have $\varphi_Y(\psi_Y(\mathcal{I})) = \mathcal{I}$. Proof. If $Y$ is smooth, then $\operatorname{Cl}(Y) = \operatorname{Pic}(Y)$, so the Cox ring $\mathcal{R}(Y)$ is $\operatorname{Pic}(Y)$-graded. Now we can apply Proposition 3.1(1,2,3) from [24].
The proposition is proved. Remark 2.9. The irrelevant ideal $B \subset \mathcal{R}(Y)$ can explicitly be described if one fixes an ample divisor $D \in K$ on $Y$ (see [1], Corollary 1.6.3.6), where the subgroup $K \subset \operatorname{WDiv}(Y)$ was introduced in Definition 2.1. We are going to describe a similar correspondence for singular Mori dream spaces $Y$ such that $\operatorname{Cl}(Y) \simeq \mathbb{Z}$. The nontriviality of $\operatorname{Cl}(Y) / \operatorname{Pic}(Y)$ brings about a notion of ‘saturation’ for homogeneous ideals $I \subset \mathcal{R}(Y)$ different from $(I : B^{\infty})$. Notation 2.10. Let $Y$ be a Mori dream space such that $\operatorname{Cl}(Y) \simeq \mathbb{Z}$. For a homogeneous ideal $I \subset \mathcal{R}(Y)$ we use the following notation: Remark 2.11. Note that if $Y$ is smooth, then $I^{\infty}$ equals to the usual saturation $(I : B^{\infty})$ with respect to the irrelevant ideal $B \subset \mathcal{R}(Y)$. One can also check that any radical homogeneous ideal $I \subset \mathcal{R}(Y)$ satisfies $I = I^{\infty}$. Proposition 2.12. Let $Y$ be a Mori dream space such that $\operatorname{Cl}(Y) \simeq \mathbb{Z}$, let $\mathcal{R}(Y)$ be its Cox ring, and let $B \subset \mathcal{R}(Y)$ be the irrelevant ideal. (1) Let $I = (f_1, \dots, f_m) \subset \mathcal{R}(Y)$ be a homogeneous ideal. Then the following decomposition holds: $\varphi_Y(I) = \sum_{j=1}^m \mathcal{O}_Y(-D_{f_j})$. (2) For any homogeneous ideal $I \subset \mathcal{R}(Y)$ we have $\psi_Y(\varphi_Y(I)) = I^{\infty}$. (3) For any ideal sheaf $\mathcal{I} \subseteq \mathcal{O}_Y$ we have $\varphi_Y(\psi_Y(\mathcal{I})) = \mathcal{I}$. Proof. We want to derive the statement from Proposition 3.1 in [24] as we did in the proof of Proposition 2.8. Yet we cannot apply it directly, because in our case the Cox ring $\mathcal{R}(Y)$ is $\operatorname{Cl}(Y)$-graded rather than $\operatorname{Pic}(Y)$-graded.
We will need the following considerations. Firstly, recall that $Y \simeq \operatorname{Proj}(\mathcal{R}(Y))$ (see [1], § 3.3.4). We introduce the truncated Cox ring, which is $\operatorname{Pic}(Y)$-graded:
$$
\begin{equation*}
\mathcal{R}(Y)^{[m]}=\bigoplus_{k=0}^{\infty} H^0(Y, \mathcal{O}_Y(m \cdot k)), \qquad m=[\operatorname{Cl}(Y) : \operatorname{Pic}(Y)].
\end{equation*}
\notag
$$
Recall that $\operatorname{Proj}(\mathcal{R}(Y))$ is canonically isomorphic to $\operatorname{Proj}(\mathcal{R}(Y)^{[m]})$ (see [12], Proposition 2.4.7(i)). Then any ideal sheaf on $Y$ can canonically be identified with an ideal sheaf on $\widetilde{Y} := \operatorname{Proj}(\mathcal{R}(Y)^{[m]})$, and vice versa. In particular, for any ideal $I \subset \mathcal{R}(Y)$ and its truncation $\widetilde{I} = I \cap \mathcal{R}(Y)^{[m]}$ we can identify the ideal sheaves $\phi_Y(I)$ and $\phi_{\widetilde{Y}}(\widetilde{I})$.
Secondly, let $I \subset \mathcal{R}(Y)$ be a homogeneous ideal, and $\widetilde{I} = I \cap \mathcal{R}(Y)^{[m]}$ be its image in the truncated Cox ring. It is not difficult to check that $I$ is saturated in the sense of Notation 2.10 (that is, we have $I = I^{\infty}$) if and only if $\widetilde{I}$ is saturated in the usual sense with respect to the irrelevant ideal $\widetilde{B} \subset \mathcal{R}(Y)^{[m]}$ (that is, we have $\widetilde{I} = (\widetilde{I} : \widetilde{B}^{\infty})$; for example, see the proof of Lemma 2.10 in [23]). Now we can prove the statement. (1) Consider the truncation $\widetilde{I} = I \cap \mathcal{R}(Y)^{[m]}$ of the ideal $I \subset \mathcal{R}(Y)$. If $\widetilde{I}$ is generated by $g_1, \dots, g_l \in \mathcal{R}(Y)^{[m]}$, then we have $\varphi_{\widetilde{Y}}(\widetilde{I}) = \sum_{j = 1}^l \mathcal{O}_{\widetilde{Y}}(-D_{g_j})$ by Proposition 3.1(1) in [24], where $\widetilde{Y} = \operatorname{Proj}(\mathcal{R}(Y)^{[m]})$. Then the canonical isomorphism $Y \simeq \widetilde{Y}$ implies that we also have $\varphi_Y(I) = \sum_{j = 1}^m \mathcal{O}_Y(-D_{f_j})$. (3) For any ideal sheaf $\mathcal{I} \subseteq \mathcal{O}_Y$ we can identify the ideal $\psi_{\widetilde{Y}}(\mathcal{I}) \subset \mathcal{R}(Y)^{[m]}$ with the truncation $\widetilde{\psi_Y(\mathcal{I})} = \psi_Y(\mathcal{I}) \cap \mathcal{R}(Y)^{[m]}$ of the ideal $\psi_Y(\mathcal{I})$. Then we can apply Proposition 3.1(3) from [24]. (2) The ideal $I \subset \mathcal{R}(Y)$ defines an ideal sheaf $\varphi_Y(I)$ on $Y$. After the canonical isomorphism $Y \simeq \widetilde{Y}$ we can think of $\varphi_Y(I)$ as an ideal sheaf on $\widetilde{Y}$. According to Proposition 3.1(2) in [24], the ideal sheaves on $\widetilde{Y}$ are in bijection with the saturated ideals $\widetilde{I} = (\widetilde{I} : \widetilde{B}^{\infty})$ in $\mathcal{R}(Y)^{[m]}$. But we can identify $\widetilde{I}$ with the truncation of the ideal $\psi_Y(\varphi_Y(I))$, so we obtain $\psi_Y(\varphi_Y(I)) = I^{\infty}$. The proposition is proved. Propositions 2.8 and 2.12 motivate the following definitions. Definition 2.13. Let $Y$ a Mori dream space and $I \subset \mathcal{R}(Y)$ be a homogeneous ideal. The saturation of $I$ is the ideal $(I : B^{\infty})$ if $Y$ is smooth and the ideal $I^{\infty}$ if $\operatorname{Cl}(Y) \simeq \mathbb{Z}$ (see Remark 2.11). An ideal is saturated if it is equal to its saturation. Definition 2.14. Let $X \subseteq Y$ be a closed subscheme of a Mori dream space such that $Y$ is smooth, or $\operatorname{Cl}(Y) \simeq \mathbb{Z}$. We refer to $\psi_Y(\mathcal{I}_X) \subset \mathcal{R}(Y)$ as the defining ideal of $X$. We define the homogeneous coordinate ring of $X$ as the $\operatorname{Cl}(Y)$-graded ring $\mathcal{R}_Y(X) = \mathcal{R}(Y) / \psi_Y(\mathcal{I}_X) = \oplus_{i \in \operatorname{Cl}(Y)} \mathcal{R}_Y(X)_i$. 2.1.3. Complete intersections in Mori dream spacesDefinition 2.15. Let $Y$ be an irreducible normal variety. A complete intersection of hypersurfaces $H_1, \dots, H_c \subset Y$ is a closed subscheme $X \subseteq Y$ with $\operatorname{codim}_Y(X) = c$ and the ideal sheaf equal to $\mathcal{I}_X = \mathcal{I}_{H_{1}} + \cdots +\mathcal{I}_{H_c}$. Definition 2.16 (see [24], Definition 2.2). Let $Y$ be an irreducible normal variety such that $\operatorname{Cl}(Y) \cong \mathbb{Z}^{\rho}$. A complete intersection $X \subseteq Y$ of hypersurfaces $H_{1}, \dots, H_{c}$ is strict if its ideal $\psi_Y(\mathcal{I}_X) \subset \mathcal{R}(Y)$ is generated by $c$ elements. Remark 2.17. Let $X \subset Y$ be a strict complete intersection in a Mori dream space such that $\operatorname{Cl}(Y) \simeq \mathbb{Z}$. The ideal $\psi_Y(\mathcal{I}_X)$ is generated by $\operatorname{codim}_Y(X)$ elements. Remark 2.7 implies that we can identify $X$ with $\operatorname{Proj}(\mathcal{R}(Y) / \psi_Y(\mathcal{I}_X))$. Then we have Lemma 2.18. Let $Y$ be a Mori dream space and $X \subseteq Y$ be a complete intersection of hypersurfaces $D_{f_1}, \dots, D_{f_c}$ defined by homogeneous elements $f_j \in \mathcal{R}(Y)$, and let $B = (g_1, \dots, g_s) \subset \mathcal{R}(Y)$ be the irrelevant ideal. Assume that one of the following conditions hold: Then $\psi_Y(\mathcal{I}_X) = (f_1, \dots, f_c)$ and $\operatorname{ht}(\psi_Y(\mathcal{I}_X)) = c$. In particular, $X$ is a strict complete intersection, and $(f_1, \dots, f_c)$ is a regular sequence. Proof. Put $I = (f_1, \dots, f_c) \subset \mathcal{R}(Y)$.
Assume that $Y$ is smooth and $s < \operatorname{ht}(B)$. By Proposition 2.8, (1), we can identify $\varphi_Y(I)$ with the ideal sheaf $\mathcal{I}_X$. According to Lemma 4.2 in [24], we have $\operatorname{ht}(I) = c$. Then by Lemma 4.1 in [24] it is saturated with respect to the irrelevant ideal $B$. Proposition 2.8, (2), implies that $\psi_Y(\varphi_Y(I)) = I$. Consequently, we have $\psi_Y(\mathcal{I}_X) = I$. Assume that $\operatorname{Cl}(Y) \simeq \mathbb{Z}$ and $I$ is a radical ideal. By Proposition 2.12, (1), we can identify $\varphi_Y(I)$ with the ideal sheaf $\mathcal{I}_X$. It is not hard to check that a radical homogeneous ideal $I \subset \mathcal{R}(Y)$ satisfies $I = I^{\infty}$. Proposition 2.12, (2), implies that $\psi_Y(\varphi_Y(I)) = I$. Consequently, we have $\psi_Y(\mathcal{I}_X) = I$. Remark 2.7 implies that $\operatorname{Proj}(\mathcal{R}(Y) / I) \simeq X$. From $Y \simeq \operatorname{Proj}(\mathcal{R}(Y))$ (see [1], § 3.3.4) we obtain
$$
\begin{equation*}
\operatorname{ht}(I)=\dim(\mathcal{R}(Y))-\dim(\mathcal{R}(Y) / I)= \operatorname{codim}_Y(X)=c
\end{equation*}
\notag
$$
(see Remark 2.17).
In both cases $\psi_Y(\mathcal{I}_X) = I$ is generated by $c$ elements and $\operatorname{ht}(\psi_Y(\mathcal{I}_X)) = c$. Then $(f_1, \dots, f_c)$ is a regular sequence by Proposition 1.5.11 in [4]. The lemma is proved. Remark 2.19. The assumptions of Lemma 2.18 cannot be dropped. Consider a Mori dream space $Y = \mathbb{P}^{1} \times \mathbb{P}^{1}$ with coordinates $(x_0 : x_1)$, $(y_0 : y_1)$, and the closed subscheme $X = ((0 : 1), (0 : 1))$. Its defining ideal has the form $\psi_Y(\mathcal{I}_X) = (x_0, y_0) \subset \mathcal{R}(Y) = \mathbb{C}[x_0, x_1, y_0, y_1]$. But we can also define $X$ by the nonsaturated ideal $I = (x_0, x_1 y_0)$ (see [24], Example 4.6). Consider a Mori dream space $Y = \mathbb{P}(1, 1, 3, 6)$ with coordinates $(x_0, \dots, x_3)$ and the homogeneous ideal $I = (x_0^3, x_1^3) \subset \mathcal{R}(Y) = \mathbb{C}[x_0, x_1, x_2, x_3]$. Its saturation $\psi_Y(\phi_Y(I)) = I^{\infty} = (x_0^3, x_1^3, x_0^2 x_1^2)$ cannot be generated by a regular sequence, hence $I$ defines a nonstrict complete intersection in $Y$ (see [25]). 2.1.4. Application: conjugate complete intersectionsLet $Y$ be a Mori dream space such that $\operatorname{Cl}(Y) \simeq \mathbb{Z}$, and let $X \subseteq Y$ be a strict complete intersection. Then Remark 2.17 implies that $\psi_Y(\mathcal{I}_X)$ is generated by a regular sequence of homogeneous elements. Definition 2.20. Let $X \subseteq Y$ be a strict complete intersection in a Mori dream space such that $\operatorname{Cl}(Y) \simeq \mathbb{Z}$, so that $\psi_Y(\mathcal{I}_X) \subset \mathcal{R}(Y)$ is generated by a regular sequence $(f_1, \dots, f_c)$ of homogeneous elements. Put $d_j = \deg(f_j) \in \mathbb{Z}_{> 0}$. We refer to $(d_1, \dots, d_c)$ as the multidegree of the complete intersection $X \subseteq Y$. Definition 2.21. Let $X = \operatorname{Proj}(\mathbb{C}[x_0, \dots, x_d] / I_X)$ be a projective variety, and let $\sigma \in \operatorname{Aut}(\mathbb{C} / \mathbb{Q})$. The conjugate variety of $X$ is $X^{\sigma} = \operatorname{Proj}(\mathbb{C}[x_0, \dots, x_d] / I_X^{\sigma})$, where $\operatorname{Aut}(\mathbb{C} / \mathbb{Q})$ acts on coefficients of polynomials in $\mathbb{C}[x_0, \dots, x_d]$. Lemma 2.22. Let $Y$ be a complex Mori dream space with $\operatorname{Cl}(Y) \simeq \mathbb{Z}$ which can be defined over $\mathbb{Q}$, and let $X \subseteq Y$ be a strict complete intersection of multidegree $(d_1, \dots, d_c)$. Then its conjugate $X^{\sigma}$ is also a strict complete intersection in $Y$ of the same multidegree $\mu$ for any $\sigma \in \operatorname{Aut}(\mathbb{C} / \mathbb{Q})$. Proof. By assumption $X$ is a strict complete intersection, so its ideal $\psi_Y(\mathcal{I}_X)$ in $\mathcal{R}(Y)$ is generated by a regular sequence of homogeneous elements $(f_1, \dots, f_c)$, where $\deg(f_j) = d_j$ (see Remark 2.17). Note that the action of $\sigma \in \operatorname{Aut}(\mathbb{C} / \mathbb{Q})$ maps the ideal $\psi_Y(\mathcal{I}_X) \subset \mathcal{R}(Y)$ to $\psi_Y(\mathcal{I}_X)^{\sigma} = (f_1^{\sigma}, \dots, f_c^{\sigma}) \subset \mathcal{R}(Y^{\sigma}) \simeq \mathcal{R}(Y)$.
This action preserves saturatedness (see Definition 2.13) because it preserves the irrelevant ideal and homogeneous components of $\mathcal{R}(Y)$ and acts $\mathbb{Q}$-linearly on coefficients of any polynomial. Consequently, Proposition 2.12 implies that $\psi_Y(\mathcal{I}_X)^{\sigma} = \psi_Y(\mathcal{I}_{X^{\sigma}})$. It is not hard to check that the $\mathbb{Q}$-linearity of this action also preserves the property of the sequence $(f_1^{\sigma}, \dots, f_c^{\sigma})$ to be regular. The lemma is proved. 2.2. Weighted complete intersectionsHere we recall the basic properties of weighted complete intersections. We refer the reader to § 2.1 for the terminology and notations related to Cox rings. Notation 2.23. Let $\rho = (a_0, \dots, a_N)$ be a tuple of positive integers. We also put $R^{\rho} = \mathbb{C}[X_0, \dots, X_N]$, where the grading $R^{\rho} = \oplus_{n = 0}^{\infty} R_n^{\rho}$ is defined by $\deg(X_i) = a_i$. Definition 2.24. Let $\rho = (a_0, \dots, a_N)$ be a tuple of positive integers. We refer to $\mathbb{P}(\rho) = \operatorname{Proj}(R^{\rho})$ as the weighted projective space with weights $\rho$. Definition 2.25 ([14], Definition 5.11). A weighted projective space $\mathbb{P}(\rho)$, where $\rho = (a_0, \dots, a_N)$, is said to be well formed if Proposition 2.26 (see [8], § 1.3.1). Any weighted projective space is isomorphic to a well-formed one. Lemma 2.27 (see [8], § 1.4.1). Let $\mathbb{P}(\rho)$ be a well-formed weighted projective space. Then $\mathcal{R}(\mathbb{P}(\rho)) \simeq R^{\rho}$. Definition 2.28. Let $\mathbb{P}(\rho)$ be a weighted projective space. A closed subscheme $X \subseteq \mathbb{P}(\rho)$ is a weighted complete intersection of multidegree $\mu = (d_1, \dots, d_c)$ if its ideal $\psi_Y(\mathcal{I}_X) \subset \mathcal{R}(\mathbb{P}(\rho))$ is generated by a regular sequence $(f_1, \dots, f_c)$ of weighted homogeneous polynomials of degrees $\deg(f_j) = d_j \in \mathbb{Z}_{> 0}$. Definition 2.29 (see [31], Definition 2.1 and Proposition 2.3). Let $X \subseteq \mathbb{P}(\rho)$ be a weighted complete intersection and $\mathcal{R}_Y(X) = \mathcal{R}(Y) / \psi_Y(\mathcal{I}_X)$ be its homogeneous coordinate ring. We define the sheaf $\mathcal{O}_X(k)$ as the coherent sheaf on $X$ associated with the graded $\mathcal{R}_Y(X)$-module $R(k)$ whose degree $i$ part is $R(k)_i$ = $\mathcal{R}_Y(X)_{k+i}$. Definition 2.30 (see [14], Definition 6.3). A closed subscheme $X \subseteq \mathbb{P}(\rho)$ is said to be quasi-smooth if its affine cone $\operatorname{Spec}(R^{\rho} / \psi_Y(\mathcal{I}_X))$ is smooth away from the origin, where $\psi_Y(\mathcal{I}_X) \subset R^{\rho}$ is the defining ideal. Definition 2.31 (cf. [7], Definition 1.1). A closed subscheme $X \subseteq \mathbb{P}(\rho)$ is said to be well formed if $\mathbb{P}(\rho)$ is well formed, and $\operatorname{codim}_X(X \cap \operatorname{Sing}(\mathbb{P}(\rho))) \geqslant 2$. Proposition 2.32 (see [7], Proposition 8, and [27], Corollary 2.14). Let $X \subseteq \mathbb{P}(\rho)$ be a weighted complete intersection. Then the following assertions are equivalent: Example 2.33. If a weighted complete intersection is not well formed, various pathologies can arise. $\bullet$ There exists a K3 surface that can be realised as a smooth and quasi-smooth hypersurface $X$ of degree 9 in $\mathbb{P}(1, 2, 2, 3)$ but that is not well formed (see [14], Note 6.15, (ii)). The naive adjunction formula (see Proposition 2.39) would imply that $\omega_X \simeq \mathcal{O}_X(1)$, which is nonsense. $\bullet$ A general weighted hypersurface of multidegree 6 in $\mathbb{P}(2, 3, 5^{(t)})$ is not well formed or quasi-smooth for any $t > 0$; nonetheless, it is smooth (see [29], Example 2.9). Lemma 2.34 ([26], Corollary 3.3). Let $X \subseteq \mathbb{P}(\rho)$ be a quasi-smooth well-formed weighted complete intersection. Then the restriction map is surjective for any ${m \in \mathbb{Z}_{\geqslant 0}}$: Definition 2.35 ([14], Definition 6.5). Let $X \subseteq \mathbb{P}(a_0, \dots, a_N)$ be a weighted complete intersection of multidegree $(d_1, \dots, d_c)$. We refer to $X$ as $\bullet$ an intersection with a linear cone if $a_i = d_j$ for some $i$ and $j$; $\bullet$ degenerate if $d_j = 1$ for some $j=1,\dots,c$. Lemma 2.36 (see [8], Theorem 3.4.4, and [14], Lemma 7.1). Let $X \subseteq \mathbb{P}(\rho)$ be a quasi-smooth well-formed weighted complete intersection which is nondegenerate. Put $\rho = (a_0, \dots, a_N)$. Then the following identities hold: Remark 2.37. Not to be an intersection with a linear cone is a rather mild restriction, provided that $X \subseteq \mathbb{P}(\rho)$ is sufficiently general (see [28], Proposition 2.9). Proposition 2.38 (see [22], Remark 4.2, and [30], Proposition 2.3). Let ${X \!\!\subset\! \mathbb{P}(\rho)}$ be a quasi-smooth well-formed weighted complete intersection of dimension at least 3. Then $\operatorname{Cl}(X)$ is generated by the class of the divisorial sheaf $\mathcal{O}_X(1)$. Proposition 2.39 (see [8], Theorem 3.3.4, [14], § 6.14, and [31], Corollary 2.6). Let $X \subset \mathbb{P}(a_0, \dots, a_N)$ be a quasi-smooth well-formed weighted complete intersection of dimension at least 2 and multidegree $(d_1, \dots, d_c)$ and $\omega_X$ be the dualising sheaf. Then the following identity holds: $\omega_X \simeq \mathcal{O}_X(\sum_{j = 1}^c d_j - \sum_{i = 0}^N a_i)$. § 3. Proof of Theorems 1.7 and 1.8In this section we prove Theorems 1.7 and 1.8. To do this we introduce the following class of varieties which behave like smooth complete intersections in $\mathbb{P}^n$. Definition 3.1. Let $Y$ be a complex Mori dream space with $\operatorname{Cl}(Y) \simeq \mathbb{Z}$ which can be defined over $\mathbb{Q}$ (see § 2.1). A smooth complete intersection $X \subseteq Y$ of multidegree $(d_1, \dots, d_c)$ is admissible if $\bullet$ $\operatorname{Pic}(X) \simeq \mathbb{Z}$ and is generated by the restriction $\mathcal{H}|_X$ of the ample generator $\mathcal{H} \in \operatorname{Cl}(Y)$, provided that $\dim(X) > 2$ (‘Lefschetz ’s hyperplane section theorem’); $\bullet$ the adjunction formula $\omega_X \simeq \omega_Y|_X \otimes \mathcal{H}|_X^{\otimes \sum d_j}$ holds if $\dim(X) > 1$; $\bullet$ the restriction map $H^0(Y, \mathcal{H}^{\otimes k}) \to H^0(X, \mathcal{H}|_X^{\otimes k})$ is surjective for each $k \in \mathbb{Z}_{\geqslant 0}$, and for each divisor $D \in H^0(X, \mathcal{H}|_X^{\otimes k})$ there exists $\widetilde{X} \in H^0(Y, \mathcal{H}^{\otimes k})$ such that $X = \widetilde{X} \cap Y$ is an admissible complete intersection in $Y$; $\bullet$ for any admissible complete intersection $X \subset Y$ and each $\sigma \in \operatorname{Aut}(\mathbb{C} / \mathbb{Q})$ its conjugate $X^{\sigma}$ is also an admissible complete intersection (cf. Lemma 2.22); $\bullet$ each smooth divisor $D \in H^0(X, \mathcal{H}|_X^{\otimes k})$ of general type in a very ample linear system satisfies the infinitesimal Torelli theorem, that is, the canonical map is injective, where $\mathcal{T}_X$ is the tangent sheaf of $X$.Remark 3.2. There exist smooth varieties with very ample canonical class such that the infinitesimal Torelli theorem fails for them (see [10]). Proposition 3.3. The following smooth complete intersections are admissible: $\bullet$ complete intersections in generalised Grassmannians; $\bullet$ well-formed weighted complete intersections such that $\dim(| \mathcal{O}_X(1)|) \geqslant 2$. Proof. A generalised Grassmannian $Y$ is the quotient of a simple algebraic group $G$ by a maximal parabolic subgroup $P$. It is a smooth Fano variety of Picard rank one (see [21], § 2). Moreover, any parabolic subgroup $P \subset Y$ is conjugate to a standard parabolic subgroup (for example, see [20], Proposition 12.2), hence the variety $Y$ can be defined over $\mathbb{Q}$. Note that Lefschetz’s hyperplane section theorem clearly holds for $X \subseteq Y$. Moreover, we have the adjunction formula for any smooth complete intersection $X \subseteq Y$. Actually, its normal bundle $(\mathcal{I}_X / \mathcal{I}_X^2)^{\vee}$ is always isomorphic to $\bigoplus_{j = 1}^c \mathcal{O}_X(d_j)$, where $(d_1, \dots, d_c)$ is the multidegree of $X$, because the ideal sheaf $\mathcal{I}_X$ can be resolved by the Koszul complex. Moreover, the restriction map $H^0(Y, \mathcal{O}_Y(k)) \to H^0(X, \mathcal{O}_X(k))$ is surjective (see the proof of Lemma 2.2 in [18]). Finally, smooth complete intersections of general type in generalised Grassmannians satisfy the infinitesimal Torelli theorem by [18], Theorem 2.6.
Weighted projective spaces $\mathbb{P}(\rho)$ are Mori dream spaces with $\operatorname{Cl}(\mathbb{P}(\rho)) \simeq \mathbb{Z}$ and can be defined over $\mathbb{Q}$. Lefschetz’s hyperplane section theorem and the adjunction formula hold for each smooth well-formed weighted complete intersection ${X \subset \mathbb{P}(\rho)}$ (see Propositions 2.38 and 2.39). The restriction map $H^0(\mathbb{P}(\rho), \mathcal{O}_{\mathbb{P}(\rho)}(k)) \to H^0(X, \mathcal{O}_X(k))$ is surjective (see Lemma 2.34). Moreover, Proposition 2.32 states that a weighted complete intersection is smooth and well formed if and only if it is quasi-smooth and does not intersect the singular locus of $\mathbb{P}(\rho)$. Consequently, a smooth divisor on a smooth well-formed weighted complete intersection $X$ is again a smooth well-formed weighted complete intersection, and each conjugate weighted complete intersection $X^{\sigma}$, where $\sigma \in \operatorname{Aut}(\mathbb{C} / \mathbb{Q})$, is also smooth and well formed (cf. Lemma 2.22). Finally, each smooth well-formed weighted complete intersection of general type $W \subseteq \mathbb{P}(\rho)$ such that $\dim(|\mathcal{O}_W(1)|) \geqslant 2$ satisfies the infinitesimal Torelli theorem by Lemma 2.36 and Theorem 2.1 in [37]. Note that the linear system $|\mathcal{O}_X(1)|$ is very ample if and only if $\mathbb{P}(\rho) = \mathbb{P}^n$. Consequently, if $\mathbb{P}(\rho) \neq \mathbb{P}^n$, then a smooth divisor $D \subset X$ lying in a very ample linear system satisfies $D \notin |\mathcal{O}_X(1)|$, hence $\dim(|\mathcal{O}_X(1)|) = \dim(|\mathcal{O}_D(1)|)$ (see Lemma 2.36). In other words, we are able to apply Usui’s infinitesimal Torelli theorem. The proposition is proved. Question 3.4. The only reason we assume that $\dim(|\mathcal{O}_X(1)|) \geqslant 2$ is because it is crucial for Usui’s proof of the infinitesimal Torelli theorem (see [37] and also [19]). Let us briefly recall its strategy. Let $X \subset \mathbb{P}(a_0, \dots, a_N)$ be a smooth well-formed weighted complete intersection of general type defined by a regular sequence $(f_1, \dots, f_c)$ of weighted homogeneous polynomials of degree $d_j$. We consider the Jacobi ring of $X$:
$$
\begin{equation*}
\begin{gathered} \, R=\mathbb{C}[x_0, \dots, x_N; y_1, \dots, y_c]/(\partial_{x_0}(F), \dots, \partial_{x_N}(F), \partial_{y_1}(F), \dots, \partial_{y_c}(F)), \\ F=\sum_{j=1}^c y_j f_j. \end{gathered}
\end{equation*}
\notag
$$
We denote by $R_{(i, j)}$ its homogeneous components under the bi-grading ${\deg(x_i) = (0, 1)}$ and $\deg(y_j) = (1, -d_j)$. Then the infinitesimal Torelli map
$$
\begin{equation*}
\Psi \colon H^1(X, \mathcal{T}_X) \to \bigoplus_{p+q=\dim(X)} \operatorname{Hom}(H^p(X, \Omega_X^q), H^{p+1}(X, \Omega_X^{q-1}))
\end{equation*}
\notag
$$
can be interpreted as follows:
$$
\begin{equation*}
\begin{gathered} \, H^1(X, \mathcal{T}_X) \simeq R_{(1, 0)}, \qquad H^p(X, \Omega_X^q) \simeq \operatorname{Hom}(R_{(q, -S)}, \mathbb{C}), \qquad S=\sum_{i=0}^N a_i-\sum_{j=1}^c d_j, \\ \begin{aligned} \, &\Psi_q \colon R_{(1, 0)} \to \operatorname{Hom}(\operatorname{Hom}(R_{(q, -S)}, \mathbb{C}), \operatorname{Hom}(R_{(q-1, -S)}, \mathbb{C})) \\ &\qquad=\operatorname{Hom}(R_{(q-1, -S)}, R_{(q, -S)}). \end{aligned} \end{gathered}
\end{equation*}
\notag
$$
In other words, $\Psi$ takes any $r \in R_{1, 0}$ to the multiplication-by-$r$ map, and the infinitesimal Torelli theorem holds precisely when $\Psi$ is nondegenerate. Usui proved this under the assumption $\dim(|\mathcal{O}_X(1)|) \geqslant 2$. Can this assumption be dropped? Definition 3.5. Let $X$ be a smooth projective variety. A Lefschetz fibration $f \colon Y = \operatorname{Bl}_{\operatorname{Bs}(\mathcal{L})}(X) \to \mathbb{P}^{1}$ over $X$ is a resolution of indeterminacy of the rational map $\Phi_{\mathcal{L}} \colon X \dashrightarrow \mathbb{P}^1$ defined by a Lefschetz pencil $\mathcal{L}$ by a single blow up. The morphism $f$ is proper and flat (see [6], Exposé XVII). A Lefschetz fibration $f$ is rigid if it is rigid as a flat family over $\mathbb{P}^1$. Equivalently, the corestriction $f|_{\mathbb{P}^{1}(\mathbb{C}) \setminus \operatorname{Crit}(f)}$ is a rigid smooth family (see [13], Proposition III.9.8). Lemma 3.6. Let $\varnothing \neq B \subseteq \mathbb{P}^1(\mathbb{C})$ be a Zariski-open subset and $h \in \mathbb{Z}[T]$ be an integer polynomial. Let $\mathcal{X}_{B, h}$ denote the set of isomorphism classes over $\mathbb{P}^1$ of all Lefschetz fibrations $f \colon Y = \operatorname{Bl}_{\operatorname{Bs}(\mathcal{L})}(X) \to \mathbb{P}^1$ over all smooth projective varieties $X$ such that the corestriction $f|_{\mathbb{P}^{1}(\mathbb{C}) \setminus \operatorname{Crit}(f)}$ is a rigid smooth family of canonically polarised varieties with Hilbert polynomial $h$, and the morphism $f$ is smooth over $B$, that is, $B \subseteq \mathbb{P}^1(\mathbb{C}) \setminus \operatorname{Crit}(f)$. Then the set $\mathcal{X}_{B, h}$ is finite. This follows from a result of Kovács–Lieblich (see [16], Theorem 2.2). Corollary 3.7. Let $\varnothing \neq B \subseteq \mathbb{P}^1(\mathbb{C})$ be a Zariski-open subset and $h \in \mathbb{Z}[T]$ be an integer polynomial. Let $\mathcal{Y}_{B, h, n}$ denote the set of isomorphism classes of smooth projective varieties of general type $X \subseteq \mathbb{P}^{n}$ admitting a Lefschetz fibration $f \colon {Y \to \mathbb{P}^{1}}$ over $X$ such that the corestriction $f|_{\mathbb{P}^{1}(\mathbb{C}) \setminus \operatorname{Crit}(f)}$ is a rigid smooth family of canonically polarised varieties with Hilbert polynomial $h$, and the morphism $f$ is smooth over $B$, that is, $B \subseteq \mathbb{P}^1(\mathbb{C}) \setminus \operatorname{Crit}(f)$. Then the set $\mathcal{Y}_{B, h, n}$ is finite. This follows from Lemma 3.6 and a Tsai-type theorem for bounded families (see [15], Theorem 2.2). Lemma 3.8. Let $X$ be a smooth projective variety and $f \colon Y \to \mathbb{P}^1$ be a Lefschetz fibration over $X$. Assume that fibres of the corestriction $f|_B$ to $B = \mathbb{P}^1(\mathbb{C}) \setminus \operatorname{Crit}(f)$ satisfy the infinitesimal Torelli theorem. Then $f|_B$ is a rigid smooth family (that is, $f \colon Y \to \mathbb{P}^1$ is a rigid Lefschetz fibration). This follows verbatim the proof of Theorem 2.5 in [15]. Lemma 3.9. Let $X$ be a smooth complex projective variety and $f \colon Y \to \mathbb{P}^1$ be a Lefschetz fibration over $X$. Put $B \!=\! \mathbb{P}^1(\mathbb{C}) \setminus \operatorname{Crit}(f)$. Assume that ${\operatorname{Crit}(f) \!\subseteq\! \mathbb{P}^1(\overline{\mathbb{Q}})}$, and the orbit of the morphism $f^{\sigma} \colon Y^{\sigma} \to \mathbb{P}^1$ under the action of $\operatorname{Stab}_{\operatorname{Aut}(\mathbb{C} / \mathbb{Q})}(B^{\sigma})$ is finite for each $\sigma \in \operatorname{Aut}(\mathbb{C} / \mathbb{Q})$. Then the morphism $f \colon Y \to \mathbb{P}^1$ can be defined over $\overline{\mathbb{Q}}$. Proof. González-Diez’s weak version of Weil’s criterion of rationality (see [11], pp. 60, 61) states that our statement is equivalent to the following one: the set of conjugates $\{f^{\sigma} \colon Y^{\sigma} \to \mathbb{P}^1\}$ under the $\operatorname{Aut}(\mathbb{C} / \mathbb{Q})$-action is finite. In turn, this follows from our assumptions and Proposition III.9.8 in [13]. Theorem 3.10 ([15], Theorem 4.2). Let $Y$ be a complex Mori dream space with $\operatorname{Cl}(Y) \simeq \mathbb{Z}$ which can be defined over $\mathbb{Q}$, and $X \subseteq Y$ be a smooth complex admissible complete intersection (see Definition 3.1) of general type of dimension at least 3. Then the following assertions are equivalent. (1) The variety $X$ can be defined over $\overline{\mathbb{Q}}$. (2) There exists a Lefschetz pencil $f \colon X \dashrightarrow \mathbb{P}^1$ with $\operatorname{Crit}(f) \subset \mathbb{P}^1(\overline{\mathbb{Q}})$. (3) There exists a Lefschetz function $f \colon X \dashrightarrow \mathbb{P}^1$ with $\operatorname{Crit}(f) \subset \mathbb{P}^1(\overline{\mathbb{Q}})$. (4) There exists a Lefschetz function $f \colon X \dashrightarrow \mathbb{P}^1$ with $|\operatorname{Crit}(f)| \leqslant 3$. (5) There exists a Lefschetz fibration $\widetilde{f} \colon Y \to \mathbb{P}^1$ over $X$ with $\operatorname{Crit}(\widetilde{f}) \subset \mathbb{P}^1(\overline{\mathbb{Q}})$. Proof. The implication $(1)\Rightarrow(2)$ follows from the existence of a Lefschetz pencil on a projective variety over $\overline{\mathbb{Q}}$. The implications $(2)\Rightarrow(3)$ and $(4)\Rightarrow(5)$ hold by definition. The implication $(3)\Rightarrow(4)$ follows from Belyi’s algorithm (see [3]). Let us prove the implication $(5)\Rightarrow(1)$.
Assume that there exists a Lefschetz fibration $f \colon Y \to \mathbb{P}^1$ over $X$ whose critical points lie in $\mathbb{P}^1(\overline{\mathbb{Q}})$. Put $B = \mathbb{P}^1(\mathbb{C}) \setminus \operatorname{Crit}(f)$. By definition a Lefschetz pencil is contained in a very ample linear system. We have assumed $X \subset Y$ to be an admissible complete intersection, hence the corestriction $f|_B$ is a smooth family of canonically polarised admissible complete intersections in $Y$. Actually, fibres of $f|_B$ are admissible complete intersections too, and we only have to apply the adjunction formula. Moreover, by the same assumption fibres of $f|_B$ satisfy the infinitesimal Torelli theorem. Then $f|_B$ is a rigid family by Lemma 3.8. Recall that the $\operatorname{Aut}(\mathbb{C} / \mathbb{Q})$-action preserves the multidegree of a complete intersection by Lemma 2.22. Consequently, the Hilbert polynomial of $X^{\sigma}$ does not depend on $\sigma \in \operatorname{Aut}(\mathbb{C} / \mathbb{Q})$ either, since all such $X^{\sigma}$ are deformationally equivalent. Then we can apply Corollary 3.7 to conclude that the number of conjugates of $f$ under the action of $\operatorname{Stab}_{\operatorname{Aut}(\mathbb{C} / \mathbb{Q})}(B^{\sigma})$ is finite for each $\sigma \in \operatorname{Aut}(\mathbb{C} / \mathbb{Q})$. So the morphism $f \colon Y \to \mathbb{P}^1$ and the variety $Y$ can be defined over $\overline{\mathbb{Q}}$ by Lemma 3.9. Finally, this implies that $X$ itself can be defined over $\overline{\mathbb{Q}}$ (see [15], Lemma 3.3). The theorem is proved. Proof of Theorems 1.7 and 1.8. Assume that one of the following holds:
$\bullet$ $X \subset Y$ is a smooth complex complete intersection of general type of dimension at least 3 in a generalised Grassmannian $Y$; $\bullet$ $X \subset \mathbb{P}(\rho)$ is a smooth well-formed complex weighted complete intersection of general type of dimension at least 3, and $\dim(|\mathcal{O}_X(1)|) \geqslant 2$. Proposition 3.3 implies that $X$ is an admissible complete intersection (in the sense of Definition 3.1) of general type. Then we can apply Theorem 3.10. § 4. A possible generalisationAs we saw in the previous section, Belyi’s theorem holds for smooth (well-formed) complete intersections of general type in a generalised Grassmannian or a weighted projective space. Moreover, we proved Belyi’s theorem for any admissible smooth complete intersection of general type in a complex Mori dream space $Y$ such that $\operatorname{Cl}(Y) \simeq \mathbb{Z}$ which can be defined over $\mathbb{Q}$. It is natural to ask whether this setting generalises to weighted generalised Grassmannians. Definition 4.1 (see [5] and [33]). A Mori dream space $Y$ such that $\operatorname{Cl}(Y) \simeq \mathbb{Z}$ is a weighted generalised Grassmannian if its Cox ring $\mathcal{R}(Y)$ admits homogeneous generators such that the associated ideal of relations coincides with the ideal of relations of the Cox ring of a usual generalised Grassmannian. In other words, a weighted generalised Grassmannian is a closed subvariety of a weighted projective space defined by the relations of a usual generalised Grassmannian in its Cox ring (see [1], § 3.2.3, for an explicit description). Hence $Y$ is a Mori dream space with $\operatorname{Cl}(Y) \simeq \mathbb{Z}$ and can be defined over $\mathbb{Q}$. Weighted Grassmannians in the sense of [5] are a special case of weighted generalised Grassmannians. We refer the reader to [34]–[36] for explicit examples. Weighted projective spaces are a special case of weighted generalised Grassmannians, hence we have to introduce the notion of well-formedness (cf. § 2.2). Definition 4.2 (cf. Definition 2.31). Let $Y \subset \mathbb{P}(\rho)$ be a weighted generalised Grassmannian. It is well formed if $\operatorname{codim}_Y(Y \cap \operatorname{Sing}(\mathbb{P}(\rho))) \geqslant 2$. A closed subscheme $X \subseteq Y$ is well formed if $Y \subset \mathbb{P}(\rho)$ is well formed, and $\operatorname{codim}_X(X \cap \operatorname{Sing}(Y)) \geqslant 2$. Theorems 1.7 and 1.8 lead to the following natural question. Question 4.3. Are smooth well-formed complete intersections in weighted generalised Grassmannians admissible in the sense of Definition 3.1? Corollary 4.4 (see Theorem 3.10). Assume that Question 4.3 has an affirmative answer. Let $Y$ be a weighted generalised Grassmannian and $X \subset Y$ be a smooth well-formed complex complete intersection of general type of dimension at least 3. Then the variety $X$ can be defined over $\overline{\mathbb{Q}}$ if and only if there exists a Lefschetz function $X \dashrightarrow \mathbb{P}^1$ with at most three critical points. Example 4.5. Let $V = \langle e_0, e_1, e_2, e_3 \rangle$ and $W = \wedge^2(V) = \langle e_{0, 1}, \dots, e_{2, 3} \rangle$ be vector spaces, and $(a_0, a_1, a_2, a_3)$ be a tuple of integers such that $a_i + a_j > a_0$ for any $0 \leqslant i < j \leqslant 3$. If we denote the coordinates on $W$ by $T_{i, j}$, then the weighted projective space $\mathbb{P} = \mathbb{P}(a_{0, 1}, a_{0,2}, a_{0,3}, a_{1,2}, a_{1,3}, a_{2, 3})$, where contains the weighted hypersurface $Y \subset \mathbb{P}$ given by the Plücker relation which is quasi-homogeneous of degree $-a_0 + a_1 + a_2 + a_3$. Assume that $Y$ is well formed; then the adjunction formula implies that
$$
\begin{equation*}
\deg(-K_{\mathbb{P}})=3 (-a_0+a_1+a_2+a_3) \quad\text{and}\quad \deg(-K_Y)=2 (-a_0+a_1+a_2+a_3).
\end{equation*}
\notag
$$
Now let $\widetilde{X} \subset \mathbb{P}$ be a general hypersurface of degree $d$ such that $X = Y \cap \widetilde{X} \subset \mathbb{P}$ is a smooth well-formed complete intersection of multidegree $(d, -a_0 + a_1 + a_2 + a_3)$. We can think of $X$ as a smooth well-formed hypersurface $X \subset Y$ of degree $d$ whose degree of the anticanonical class is equal to
$$
\begin{equation*}
\deg(-K_X)=\deg(-K_{\mathbb{P}})-(-a_0+a_1+a_2+a_3)-d=\deg(-K_Y)-d.
\end{equation*}
\notag
$$
If $d > -a_0 + a_1 + a_2 + a_3$, then $X$ is of general type, so we can apply Theorem 1.8.
Citation in format AMSBIB
\Bibitem{Ovc25} |
|
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|
Contact us: |
Terms of Use
|
Registration to the website |
Logotypes |
|