Automorphisms of weighted complete intersections

TL;DR

This paper proves that smooth, well-formed weighted complete intersections generally have finite automorphism groups, with some clearly identified exceptions, advancing understanding of their symmetry properties.

Contribution

It establishes the finiteness of automorphism groups for a broad class of weighted complete intersections, identifying specific cases where this does not hold.

Findings

Finite automorphism groups for smooth well-formed weighted complete intersections

Identification of obvious exceptions to finiteness

Contribution to the classification of symmetries in algebraic geometry

Abstract

We show that smooth well formed weighted complete intersections have finite automorphism groups, with several obvious exceptions.

Tables

Table 1. Table 1. Fano weighted complete intersections in dimensions 1 1 1 and 2 2 2

No.	$\mathbb{P}$	Degrees
dimension $1$
1.1	${\mathbb{P}}^2$	$2$
1.2	${\mathbb{P}}^1$	$\varnothing$
dimension $2$
2.1	$\mathbb{P}(1^2,2,3)$	$6$
2.2	$\mathbb{P}(1^3,2)$	$4$
2.3	${\mathbb{P}}^3$	$3$
2.4	${\mathbb{P}}^4$	$2,2$
2.5	${\mathbb{P}}^3$	$2$
2.6	${\mathbb{P}}^2$	$\varnothing$

Table 2. Table 2. Calabi–Yau weighted complete intersections in dimensions 1 1 1 and 2 2 2

No.	$\mathbb{P}$	Degrees
dimension $1$
1.1	$\mathbb{P}(1,2,3)$	$6$
1.2	$\mathbb{P}(1,1,2)$	$4$
1.3	${\mathbb{P}}^2$	$3$
1.4	${\mathbb{P}}^3$	$2,2$
dimension $2$
2.1	$\mathbb{P}(1^3,3)$	$6$
2.2	${\mathbb{P}}^3$	$4$
2.3	${\mathbb{P}}^4$	$2,3$
2.4	${\mathbb{P}}^5$	$2,2,2$

Equations

$$\mathbb{P}=\mathbb{P}(a_0,\dots ,a_N)=\mathrm{Proj}\mathbb{k}[x_0,\dots ,x_N].$$

$${\mathrm{codim}}_X\left(X\cap \mathrm{Sing}\mathbb{P}\right)⩾2,$$

$$i_X=\sum a_j-\sum d_i.$$

$${\omega }_X={\mathcal{O}}_X\left(-i_X\right).$$

$$p_{\ast }({\mathcal{O}}_{C_X^{\ast }})=\underset{m\in \mathbb{Z}}{⨁}{\mathcal{O}}_X(m).$$

$$U=\big{(}\mathrm{Spec}\,R\big{)}\setminus\{0\}\cong\mathbb{A}^{N+1}\setminus\{0\}.$$

$$Y={\mathrm{Spec}}_{\mathbb{P}}\left(\underset{m⩾0}{⨁}{\mathcal{O}}_{\mathbb{P}}(m)\right)$$

$$Y\to \mathrm{Spec}R\cong {\mathbb{A}}^{N+1}$$

$$U\cong {\mathrm{Spec}}_{\mathbb{P}}\left(\underset{m\in \mathbb{Z}}{⨁}{\mathcal{O}}_{\mathbb{P}}(m)\right).$$

$$C_X^{\ast }\cong X{\times }_{\mathbb{P}}U.$$

$$C_X^{\ast }\cong {\mathrm{Spec}}_X\left(\underset{m\in \mathbb{Z}}{⨁}{\mathcal{O}}_X(m)\right),$$

$$P_X(t)=\sum _{m⩾0}{h}^0(X,{\mathcal{O}}_X(m))t^m.$$

$$P_X(t)=\frac{{\prod }_{j=1}^k(1-t^{d_j})}{{\prod }_{i=0}^N(1-t^{a_i})}.$$

$$\sum _{m⩾0}\dim (S_m)t^m=\frac{{\prod }_{j=1}^k(1-t^{d_j})}{{\prod }_{i=0}^N(1-t^{a_i})},$$

$$\mathrm{Spec}S\subset \mathrm{Spec}R\cong {\mathbb{A}}^{N+1}$$

$$S\cong H^0\left(C_X,{\mathcal{O}}_{C_X}\right).$$

$$\dots \to H_P^i\left(C_X,{\mathcal{O}}_{C_X}\right)\to H^i\left(C_X,{\mathcal{O}}_{C_X}\right)\to H^i\left(C_X^{\ast },{\mathcal{O}}_{C_X^{\ast }}\right)\to H_P^{i+1}\left(C_X,{\mathcal{O}}_{C_X}\right)\to \dots ,$$

$$H_P^i\left(C_X,{\mathcal{O}}_{C_X}\right)=0$$

$$H^i\left(C_X,{\mathcal{O}}_{C_X}\right)\cong H^i\left(C_X^{\ast },{\mathcal{O}}_{C_X^{\ast }}\right)$$

$$H^i(C_X^{\ast },{\mathcal{O}}_{C_X^{\ast }})\cong H^i(X,p_{\ast }{\mathcal{O}}_{C_X^{\ast }})$$

$$p_{\ast }({\mathcal{O}}_{C_X^{\ast }})=\underset{m\in \mathbb{Z}}{⨁}{\mathcal{O}}_X(m).$$

$$S\cong \underset{m⩾0}{⨁}{H}^0\left(X,{\mathcal{O}}_X(m)\right).$$

$$H^i\left(C_X,{\mathcal{O}}_{C_X}\right)\cong \underset{m\in \mathbb{Z}}{⨁}{H}^i\left(X,{\mathcal{O}}_X(m)\right).$$

$$H^{0}\big{(}\mathbb{P},\mathcal{O}_{\mathbb{P}}(m)\big{)}\to H^{0}\big{(}X,\mathcal{O}_{X}(m)\big{)}$$

$$\frac{{\prod }_{j=1}^k(1-t^{d_j})}{{\prod }_{i=0}^N(1-t^{a_i})}=\frac{{\prod }_{j^{\prime }=1}^{k^{\prime }}(1-t^{d_{j^{\prime }}^{\prime }})}{{\prod }_{i^{\prime }=0}^{N^{\prime }}(1-t^{a_{i^{\prime }}^{\prime }})}$$

$$\frac{{\prod }_{j=1}^k(1-t^{d_j})}{{\prod }_{i=0}^N(1-t^{a_i})}=P_X(t)=P_{X^{\prime }}(t)=\frac{{\prod }_{j^{\prime }=1}^{k^{\prime }}(1-t^{d_{j^{\prime }}^{\prime }})}{{\prod }_{i^{\prime }=0}^{N^{\prime }}(1-t^{a_{i^{\prime }}^{\prime }})}.$$

$$H^{4}\big{(}\mathrm{Gr}(2,5),\mathbb{Z}\big{)}\cong\mathbb{Z}^{2}.$$

$$H^4(X,\mathbb{Z})\cong {\mathbb{Z}}^2.$$

$${\mathrm{Aut}}_{\mathbb{P}}(X)\cong \mathrm{Aut}(\mathbb{P};X)/\mathrm{Aut}(\mathbb{P};X{)}_{\mathrm{id}}$$

$$H^{n}\big{(}X,\Omega^{1}_{X}\otimes\mathcal{O}_{X}(-i)\big{)}=0$$

Full text

Victor Przyjalkowski was partially supported by Laboratory of Mirror Symmetry NRU HSE, RF Government grant, ag. № 14.641.31.0001. Constantin Shramov was supported by the Russian Academic Excellence Project “5-100” and by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS”. Both authors are Young Russian Mathematics award winners and would like to thank its sponsors and jury.

1. Introduction

Studying algebraic varieties, it is important to understand their automorphism groups. In some particular cases these groups have especially nice structure. For instance, recall the following classical result due to H. Matsumura and P. Monsky (cf. [KS58, Lemma 14.2]).

Theorem 1.1** (see [MaMo63, Theorems 1 and 2]).**

Let $X\subset\mathbb{P}^{N}$, $N\geqslant 3$, be a smooth hypersurface of degree $d\geqslant 3$. Suppose that $(N,d)\neq(3,4)$. Then the group $\mathrm{Aut}(X)$ is finite.

The following beautiful generalization of Theorem 1.1 was proved by O. Benoist.

Theorem 1.2** ([Be13, Theorem 3.1]).**

Let $X$ be a smooth complete intersection of dimension at least $2$ in $\mathbb{P}^{N}$ that is not contained in a hyperlplane. Suppose that $X$ does not coincide with $\mathbb{P}^{N}$, is not a quadric hypersurface in $\mathbb{P}^{N}$, and is not a $K3$ surface. Then the group $\mathrm{Aut}(X)$ is finite.

The goal of this paper is to generalize Theorems 1.1 and 1.2 to the case of smooth weighted complete intersections. We refer the reader to [Do82] and [IF00] (or to §2 below) for definitions and basic properties of weighted projective spaces and complete intersections therein. Our main result is as follows.

Theorem 1.3**.**

Let $X$ be a smooth well formed weighted complete intersection of dimension $n$. Suppose that either $n\geqslant 3$, or $K_{X}\neq 0$. Then the group $\mathrm{Aut}(X)$ is finite unless $X$ is isomorphic either to $\mathbb{P}^{n}$ or to a quadric hypersurface in $\mathbb{P}^{n+1}$.

Under a minor additional assumption (cf. Definition 2.2 below) one can make the assertion of Theorem 1.3 more precise.

Corollary 1.4**.**

Let $X\subset\mathbb{P}$ be a smooth well formed weighted complete intersection of dimension $n$ that is not an intersection with a linear cone. Suppose that either $n\geqslant 3$, or $K_{X}\neq 0$. Then the group $\mathrm{Aut}(X)$ is finite unless $X=\mathbb{P}\cong\mathbb{P}^{n}$ or $X$ is a quadric hypersurface in $\mathbb{P}\cong\mathbb{P}^{n+1}$.

Note that if $X$ is not an intersection with a linear cone, then the assumption of Theorem 1.3 is equivalent to the requirement that $X$ is not one of the weighted complete intersections listed in Table LABEL:table:K3 below.

We refer the reader to [HMX13], [KPS18, Theorem 1.1.2], and [CPS19, Theorem 1.2] for other results concerning finiteness of automorphism groups.

Theorem 1.3 is mostly implied by the results of [Fle81] (see Theorem 4.5 below). However, some cases are not covered by [Fle81] and have to be classified and treated separately (see Proposition 3.7(iv) and Lemma 4.8).

To deduce Corollary 1.4 from Theorem 1.3, we need the the following assertion that is well known to experts but for which we did not manage to find a proper reference (and which we find interesting on its own). We will say that a weighted complete intersection $X\subset\mathbb{P}=\mathbb{P}(a_{0},\ldots,a_{N})$ of multidegree $(d_{1},\ldots,d_{k})$ is normalized if the inequalities $a_{0}\leqslant\ldots\leqslant a_{N}$ and $d_{1}\leqslant\ldots\leqslant d_{k}$ hold.

Proposition 1.5** (cf. [IF00, Lemma 18.3]).**

Let $X\subset\mathbb{P}(a_{0},\ldots,a_{N})$ and $X^{\prime}\subset\mathbb{P}(a^{\prime}_{0},\ldots,a^{\prime}_{N^{\prime}})$ be normalized quasi-smooth well formed weighted complete intersections of multidegrees $(d_{1},\ldots,d_{k})$ and $(d^{\prime}_{1},\ldots,d^{\prime}_{k^{\prime}})$, respectively, such that $X$ and $X^{\prime}$ are not intersections with linear cones. Suppose that $X\cong X^{\prime}$ and $\dim X\geqslant 3$. Then $N=N^{\prime}$, $k=k^{\prime}$, $a_{i}=a^{\prime}_{i}$ for every $0\leqslant i\leqslant N$, and $d_{j}=d^{\prime}_{j}$ for every $1\leqslant j\leqslant k$.

When the first draft of this paper was completed, A. Massarenti informed us that a result essentially similar to Theorem 1.3 was proved earlier in [ACM18, Proposition 5.7]. Note however that in [ACM18, §5] the authors work with smooth weighted complete intersections subject to certain strong additional assumptions (see [ACM18, Assumptions 5.2] for details).

Throughout the paper we work over an algebraically closed field $\Bbbk$ of characteristic zero.

The plan of our paper is as follows. In §2 we recall some preliminary facts on weighted complete intersections. In §3 we show uniqueness of presentation of a variety as a weighted complete intersection, that is, we prove Proposition 1.5. Finally, in §4 we prove Theorem 1.3 and Corollary 1.4.

We are grateful to B. Fu, B. van Geemin, A. Kuznetsov, A. Massarenti, and T. Sano for useful discussions. We also thank the referee for his suggestions regarding the first draft of the paper.

2. Preliminaries

In this section we recall the basic properties of weighted complete intersections. We refer the reader to [Do82] and [IF00] for more details. Some properties of smooth weighted complete intersections can be also found in the earlier paper [Mo75].

Let $a_{0},\ldots,a_{N}$ be positive integers. Consider the graded algebra $\Bbbk[x_{0},\ldots,x_{N}]$, where the grading is defined by assigning the weights $a_{i}$ to the variables $x_{i}$. Put

[TABLE]

The weighted projective space $\mathbb{P}$ is said to be well formed if the greatest common divisor of any $N$ of the weights $a_{i}$ is $1$. Every weighted projective space is isomorphic to a well formed one, see [Do82, 1.3.1]. A subvariety $X\subset\mathbb{P}$ is said to be well formed if $\mathbb{P}$ is well formed and

[TABLE]

where the dimension of the empty set is defined to be $-1$.

We say that a subvariety $X\subset\mathbb{P}$ of codimension $k$ is a weighted complete intersection of multidegree $(d_{1},\ldots,d_{k})$ if its weighted homogeneous ideal in $\Bbbk[x_{0},\ldots,x_{N}]$ is generated by a regular sequence of $k$ homogeneous elements of degrees $d_{1},\ldots,d_{k}$. This is equivalent to the requirement that the codimension of (every irreducible component of) the variety $X$ equals the (minimal) number of generators of the weighted homogeneous ideal of $X$, cf. [Ha77, Theorem II.8.21A(c)]. Note that $\mathbb{P}$ can be thought of as a weighted complete intersection of codimension [math] in itself; this gives us a smooth Fano variety if and only if $\mathbb{P}\cong\mathbb{P}^{N}$.

Definition 2.1** (see [IF00, Definition 6.3]).**

Let $p\colon\mathbb{A}^{N+1}\setminus\{0\}\to\mathbb{P}$ be the natural projection to the weighted projective space. A subvariety $X\subset\mathbb{P}$ is called quasi-smooth if $p^{-1}(X)$ is smooth.

Note that a smooth well formed weighted complete intersection is always quasi-smooth, see [PSh16, Corollary 2.14].

The following definition describes weighted complete intersections that are to a certain extent analogous to complete intersections in a usual projective space that are contained in a hyperplane.

Definition 2.2** (cf. [IF00, Definition 6.5]).**

A weighted complete intersection $X\subset\mathbb{P}$ is said to be an intersection with a linear cone if one has $d_{j}=a_{i}$ for some $i$ and $j$.

Remark 2.3*.*

A general quasi-smooth well formed weighted complete intersection is isomorphic to a quasi-smooth well formed weighted complete intersection that is not an intersection with a linear cone, cf. [PSh16, Remark 5.2]. Note however that this does not hold without the generality assumption. For instance, a general weighted complete intersection of bidegree $(2,4)$ in $\mathbb{P}(1^{n+2},2)$ is isomorphic to a quartic hypersurface in $\mathbb{P}^{n+1}$, while certain weighted complete intersections of this type are isomorphic to double covers of an $n$-dimensional quadric branched over an intersection with a quartic.

Given a subvariety $X\subset\mathbb{P}$, we denote by $\mathcal{O}_{X}(1)$ the restriction of the sheaf $\mathcal{O}_{\mathbb{P}}(1)$ to $X$, see [Do82, 1.4.1]. Note that the sheaf $\mathcal{O}_{\mathbb{P}}(1)$ may be not invertible. However, if $X$ is well formed, then $\mathcal{O}_{X}(1)$ is a well-defined divisorial sheaf on $X$. Furthermore, if $X$ is well formed and smooth, then $\mathcal{O}_{X}(1)$ is a line bundle on $X$.

Lemma 2.4** ([Ok16, Remark 4.2], [PST17, Proposition 2.3], cf. [Mo75, Theorem 3.7]).**

Let $X$ be a quasi-smooth well formed weighted complete intersection of dimension at least $3$. Then the class of the divisorial sheaf $\mathcal{O}_{X}(1)$ generates the group $\mathrm{Cl}\,(X)$ of classes of Weil divisors on $X$. In particular, under the additional assumption that $X$ is smooth, the class of the line bundle $\mathcal{O}_{X}(1)$ generates the group $\mathrm{Pic}\,(X)$.

One can describe the canonical class of a weighted complete intersection. For a weighted complete intersection $X$ of multidegree $(d_{1},\ldots,d_{k})$ in $\mathbb{P}$, define

[TABLE]

Let $\omega_{X}$ be the dualizing sheaf on $X$.

Theorem 2.5** (see [Do82, Theorem 3.3.4], [IF00, 6.14]).**

Let $X$ be a quasi-smooth well formed weighted complete intersection. Then

[TABLE]

Using the bounds on numerical invariants of smooth weighted complete intersections found in [CCC11, Theorem 1.3], [PSh16, Theorem 1.1], and [PST17, Corollary 5.3(i)], one can easily obtain the classically known lists of all smooth Fano weighted complete intersections of small dimensions. Namely, we have the following.

Lemma 2.6**.**

Let $X$ be a smooth well formed Fano weighted complete intersection of dimension at most $2$ in $\mathbb{P}$ that is not an intersection with a linear cone. Then $X$ is one of the varieties listed in Table LABEL:table:dim_12.

Remark 2.7*.*

Let $X$ be a smooth well formed Fano weighted complete intersection of dimension $2$. If we do not assume that $X$ is not an intersection with a linear cone, we cannot use the classification provided by Lemma 2.6. However, Lemma 2.6 applied together with Remark 2.3 shows that if $i_{X}=1$, then $X$ is a del Pezzo surface of (anticanonical) degree at most $4$.

Recall that the Fano index of a Fano variety $X$ is defined as the maximal integer $m$ such that the canonical class $K_{X}$ is divisible by $m$ in the Picard group of $X$. Theorem 2.5 and Lemmas 2.4 and 2.6 imply the following.

Corollary 2.8**.**

Let $X$ be a smooth Fano well formed weighted complete intersection of dimension at least $2$. Then the Fano index of $X$ equals $i_{X}$.

Proof.

Suppose that $\dim X=2$. Note that the Fano index is constant in the family of smooth weighted complete intersections of a given multidegree in a given weighted projective space. Similarly, $i_{X}$ is constant in such a family. Thus, by Remark 2.3 we may assume that $X$ is not an intersection with a linear cone. Now the assertion follows from the classification provided in Lemma 2.6.

If $\dim X\geqslant 3$, then we apply Theorem 2.5 together with Lemma 2.4. ∎

Note that the assertion of Corollary 2.8 fails in dimension $1$: if $X$ is a conic in $\mathbb{P}^{2}$, then $i_{X}=1$, while the Fano index of $X$ equals $2$.

Lemma 2.9**.**

Let $X\subset\mathbb{P}(a_{0},\ldots,a_{N})$ be a smooth well formed weighted complete intersection of multidegree $(d_{1},\ldots,d_{k})$. Then a general weighted complete intersection $X^{\prime}$ of multidegree $(d_{1},\ldots,d_{k})$ in $\mathbb{P}(1,a_{0},\ldots,a_{N})$ is smooth and well formed, and $i_{X^{\prime}}=i_{X}+1$.

Proof.

Straightforward. ∎

For the converse of Lemma 2.9, see [PST17, Theorem 1.2].

Similarly to Lemma 2.6, we can classify three-dimensional smooth well formed Fano weighted complete intersections that are not intersections with a linear cone (see [PSh18, Table 2]). This together with Lemma 2.9 allows us to classify smooth well formed weighted complete intersections of dimension up to $2$ with trivial canonical class.

Lemma 2.10**.**

Let $X$ be a smooth well formed weighted complete intersection of dimension at most $2$ in $\mathbb{P}$ that is not an intersection with a linear cone. Suppose that $K_{X}=0$. Then $X$ is one of the varieties listed in Table LABEL:table:K3.

Remark 2.11*.*

Note that each of the four families of elliptic curves listed in Table LABEL:table:K3 in fact contains all elliptic curves up to isomorphism. This is not the case for $K3$ surfaces. For instance, a general member of the family 2.2 in Table LABEL:table:K3 does not appear in the family 2.1 due to degree reasons.

3. Uniqueness of embeddings

In this section we prove Proposition 1.5. Let us start with a couple of facts that are well known and can be proved similarly to their analogs for complete intersections in the usual projective space. However, we provide their proofs for the reader’s convenience.

The proof of the following was suggested to us by A. Kuznetsov.

Lemma 3.1**.**

Let $X\subset\mathbb{P}=\mathbb{P}(a_{0},\ldots,a_{N})$ be a subvariety. Let $C_{X}^{*}$ be a complement of the affine cone over $X$ to its vertex, and let $p\colon C_{X}^{*}\to X$ be the projection. Then one has

[TABLE]

Proof.

Denote the polynomial ring $\Bbbk[x_{0},\ldots,x_{N}]$ by $R$ and let

[TABLE]

Let

[TABLE]

be the relative spectrum. Since $R\cong\Gamma\left(\bigoplus_{m\geqslant 0}\mathcal{O}_{\mathbb{P}}(m)\right)$, one obtains the map

[TABLE]

which is a weighted blow up of the origin (with weights $a_{0},\ldots,a_{N}$). Cutting out the origin one gets the isomorphism

[TABLE]

Consider the fibered product

[TABLE]

Taking into account that $\mathcal{O}_{\mathbb{P}}(m)|_{X}=\mathcal{O}_{X}(m)$ by definition, we obtain an isomorphism

[TABLE]

and the assertion of the lemma follows. ∎

For a subvariety $X\subset\mathbb{P}(a_{0},\ldots,a_{N})$, we define the Poincaré series

[TABLE]

The proof of the following fact was kindly shared with us by T. Sano.

Proposition 3.2** (see [Do82, Theorem 3.4.4], [PST17, Lemma 2.4], [Do82, Theorem 3.2.4(iii)]).**

Let $X\subset\mathbb{P}(a_{0},\ldots,a_{N})$ be a weighted complete intersection of multidegree $(d_{1},\ldots,d_{k})$. The following assertions hold.

• (i)

One has

[TABLE]

• (ii)

One has $H^{i}(X,\mathcal{O}_{X}(m))=0$ for all $m\in\mathbb{Z}$ and all $0<i<\dim X$.

Proof.

Denote the graded polynomial ring $\Bbbk[x_{0},\ldots,x_{N}]$ by $R$, denote the weighted homogeneous ideal $(f_{1},\ldots,f_{k})$ that defines $X$ by $I$, and put $S=R/I$, so that $X\cong\mathrm{Proj}\,S$. From regularity of the sequence $f_{1},\ldots,f_{k}$ it easily follows that

[TABLE]

where $S_{m}$ is the $m$-th graded component of $S$.

Denote the affine cone

[TABLE]

over $X$ by $C_{X}$. By construction, one has an isomorphism of graded algebras

[TABLE]

Following Lemma 3.1 denote the complement of $C_{X}$ to its vertex $P$ by $C^{*}_{X}$. Consider the local cohomology groups $H^{\bullet}_{P}(C_{X},\mathcal{O}_{C_{X}})$, see, for instance, [Ha67, p. 2, Definition]. We have the exact sequence

[TABLE]

see [Ha67, Corollary 1.9]. Since $C_{X}$ is a complete intersection in the affine space, it is Cohen–Macaulay. Therefore, by [Ha67, Proposition 3.7] and [Ha67, Theorem 3.8] one has

[TABLE]

for all $i<\dim C_{X}=\dim X+1$. Hence, we obtain an isomorphism

[TABLE]

for all $i<\dim X$.

Finally, denote the natural projection $C^{*}_{X}\to X$ by $p$. Then

[TABLE]

for all $i$. On the other hand, by Lemma 3.1 we have

[TABLE]

For $i=0$, we combine the isomorphisms (3.2), (3.3), (3.4), and (3.5) to obtain an isomorphism of graded algebras

[TABLE]

This together with (3.1) gives assertion (i).

For $0<i<\dim X$, we combine the isomorphisms (3.3), (3.4), and (3.5) to obtain an isomorphism

[TABLE]

Since $C_{X}$ is an affine variety, we have $H^{i}(C_{X},\mathcal{O}_{C_{X}})=0$ for all $i>0$, see for instance [Ha77, Theorem III.3.5]. This proves assertion (ii). ∎

Proposition 3.2(i) implies the following property that can be considered as an analog of linear normality for usual complete intersections.

Corollary 3.3**.**

Let $X\subset\mathbb{P}$ be a quasi-smooth well formed weighted complete intersection. Then the restriction map

[TABLE]

is surjective for every $m\in\mathbb{Z}$.

Proof.

The dimension of the image of the restriction map is computed by the coefficient in the Poincaré series (3.1). On the other hand, by Proposition 3.2(i) this coefficient also equals the dimension of $H^{0}(X,\mathcal{O}_{X}(m))$. ∎

To proceed, we will need an elementary observation.

Lemma 3.4** (see [IF00, Lemma 18.3]).**

Let $N$ and $N^{\prime}$ be positive integers, and $k$ and $k^{\prime}$ be non-negative integers. Let $a_{0}\leqslant\ldots\leqslant a_{N}$, $a_{0}^{\prime}\leqslant\ldots\leqslant a_{N^{\prime}}^{\prime}$, $d_{1}\leqslant\ldots\leqslant d_{k}$, and $d_{1}^{\prime}\leqslant\ldots\leqslant d_{k^{\prime}}^{\prime}$ be positive integers. Suppose that

[TABLE]

as rational functions in the variable $t$. Suppose that $a_{i}\neq d_{j}$ for all $i$ and $j$, and $a_{i^{\prime}}^{\prime}\neq d_{j^{\prime}}^{\prime}$ for all $i^{\prime}$ and $j^{\prime}$. Then $N=N^{\prime}$, $k=k^{\prime}$, $a_{i}=a^{\prime}_{i}$ for every $0\leqslant i\leqslant N$, and $d_{j}=d^{\prime}_{j}$ for every $1\leqslant j\leqslant k$.

Proof.

Note that numerators and denominators of the rational functions in the left and the right hand sides of (3.6) may have common divisors (for instance, if some $d_{j}$ is divisible by some $a_{i}$, or another way around). To prove the assertion, we will keep track of the numbers that are roots of either the numerator or the denominator, but not both of them.

Observe that equality (3.6) is equivalent to the equality obtained from (3.6) by interchanging the collections $\{a_{i}\}$, $\{a_{j}^{\prime}\}$ with $\{d_{s}\}$, $\{d_{r}^{\prime}\}$, respectively. Thus we may assume that $d_{k}$ is the maximal number among $a_{N}$, $a_{N^{\prime}}^{\prime}$, $d_{k}$, and $d_{k^{\prime}}^{\prime}$. By assumption we know that $a_{N}<d_{k}$. Let $\zeta$ be a primitive $d_{k}$-th root of unity. Then $\zeta$ is a root of the numerator of the left hand side of (3.6) but not the root of its denominator. Hence $\zeta$ is a root of the numerator $\nu(t)$ of the right hand side of (3.6) as well. Since $d_{j^{\prime}}^{\prime}\leqslant d_{k}$ for all $j^{\prime}$, we see that $\nu(t)$ is divisible by $1-t^{d_{k}}$. Cancelling the factor $1-t^{d_{k}}$ from (3.6), we complete the proof of the lemma by induction. ∎

Now we prove the main result of this section.

Proof of Proposition 1.5.

By Lemma 2.4, the group $\mathrm{Cl}\,(X)$ is generated by the class of the line bundle $\mathcal{O}_{X}(1)$, while the group $\mathrm{Cl}\,(X^{\prime})$ is generated by the class of the line bundle $\mathcal{O}_{X^{\prime}}(1)$. Therefore, we see from Proposition 3.2(i) that

[TABLE]

Since neither $X$ nor $X^{\prime}$ is an intersection with a linear cone, the required assertion follows from Lemma 3.4. ∎

Remark 3.5*.*

The assertion of Proposition 1.5 also holds for smooth Fano weighted complete intersections of dimension $2$. This follows from their explicit classification, see Lemma 2.6. Note however that the assertion fails in dimension $1$. Indeed, a conic in $\mathbb{P}^{2}$ is isomorphic to $\mathbb{P}^{1}$ (which can be considered as a complete intersection of codimension [math] in itself).

Remark 3.6*.*

We point out that the assumption that $X$ is Fano is essential for the validity of Proposition 1.5 in dimension $2$. For instance, there exist smooth quartics in $\mathbb{P}^{3}$ that also have a structure of a double cover of $\mathbb{P}^{2}$ branched in a sextic curve, see e.g. [MaMo63, Proof of Theorem 4]. Similarly, by Remark 2.11 the assertion of Proposition 1.5 fails for elliptic curves.

We do not know if the assertion of Proposition 1.5 holds in dimension $2$ in the case when $X$ and $X^{\prime}$ are quasi-smooth del Pezzo surfaces. We point out that the assumption that $X$ alone is quasi-smooth is not enough for this. Indeed, the weighted projective plane $\mathbb{P}(1,1,2)$ (which can be considered as a quasi-smooth well formed weighted complete intersection of codimension [math] in itself) can be embedded as a quadratic cone into $\mathbb{P}^{3}$ (which is not quasi-smooth).

In the proof of Theorem 1.3, we will need a classification of weighted complete intersections of large Fano index.

Proposition 3.7** (cf. [PSh18, Theorem 2.7]).**

Let $X\subset\mathbb{P}$ be a smooth well formed Fano weighted complete intersection of dimension $n\geqslant 2$. Then

• (i)

one has $i_{X}\leqslant n+1$;

• (ii)

if $i_{X}=n+1$, then $X\cong\mathbb{P}^{n}$;

• (iii)

if $i_{X}=n$, then $X$ is isomorphic to a quadric in $\mathbb{P}^{n+1}$;

• (iv)

if $i_{X}=n-1$ and $n\geqslant 3$, then $X$ is isomorphic to a hypersurface of degree $6$ in $\mathbb{P}=\mathbb{P}(1^{n},2,3)$, or to a hypersurface of degree $4$ in $\mathbb{P}=\mathbb{P}(1^{n+1},2)$, or to a cubic hypersurface in $\mathbb{P}=\mathbb{P}^{n+1}$, or to an intersection of two quadrics in $\mathbb{P}=\mathbb{P}^{n+2}$.

Proof.

Recall that $i_{X}$ equals the Fano index of $X$ by Corollary 2.8. By [IP99, Corollary 3.1.15], we know that $i_{X}\leqslant n+1$; if $i_{X}=n+1$, then $X$ is isomorphic to $\mathbb{P}^{n}$; and if $i_{X}=n$, then $X$ is isomorphic to a quadric in $\mathbb{P}^{n+1}$. This proves assertions (i), (ii), and (iii).

Now suppose that $i_{X}=n-1$ and $n\geqslant 3$. Note that $\mathrm{Pic}\,(X)\cong\mathbb{Z}$ by Lemma 2.4. Thus it follows from the classification of smooth Fano varieties of Fano index $n-1$ (see [Fu80-84] or [IP99, Theorem 3.2.5]) that $X$ is isomorphic either to one of the weighted complete intersections listed in assertion (iv), or to a linear section of the Grassmannian $\mathrm{Gr}(2,5)$ in its Plücker embedding.

It remains to show that a weighted complete intersection $X$ cannot be isomorphic to a linear section of the Grassmannian $\mathrm{Gr}(2,5)$. Suppose that $X$ is isomorphic to such a variety. Then $n\leqslant 6$. If $n=3$, then the Fano index of $X$ equals $2$ and its anticanonical degree is equal to $40$. If $n=4$, then the Fano index of $X$ equals $3$ and its anticanonical degree is equal to $405$. In both cases we see from Remark 2.3 that there exists a smooth weighted complete intersection $X$ with the same $n$ and $i_{X}$ that is not an intersection with a linear cone. This is impossible by a classification of smooth Fano weighted complete intersections of dimensions $3$ and $4$, see [PSh18, Table 2] (cf. [IP99, §12]) and [PSh16, Table 1], respectively.

Therefore, we see that $5\leqslant n\leqslant 6$. Recall that

[TABLE]

Thus, Lefschetz hyperplane section theorem implies that

[TABLE]

On the other hand, since $X$ is a weighted complete intersection of dimension greater than $4$, one has $H^{4}(X,\mathbb{Z})\cong\mathbb{Z}$ by the Lefschetz-type theorem for complete intersections in toric varieties, see [Ma99, Proposition 1.4]. The obtained contradiction completes the proof of assertion (iv). ∎

Proposition 1.5 allows us to prove more precise classification results concerning Fano weighted complete intersections (which we will not use directly in our further proofs).

Corollary 3.8**.**

Let $X\subset\mathbb{P}$ be a smooth well formed Fano weighted complete intersection of dimension $n\geqslant 2$ that is not an intersection with a linear cone. Then

• (i)

if $i_{X}=n+1$, then $X=\mathbb{P}=\mathbb{P}^{n}$;

• (ii)

if $i_{X}=n$, then $X$ is a quadric in $\mathbb{P}=\mathbb{P}^{n+1}$;

• (iii)

if $i_{X}=n-1$, then $X$ is either a hypersurface of degree $6$ in $\mathbb{P}=\mathbb{P}(1^{n},2,3)$, or a hypersurface of degree $4$ in $\mathbb{P}=\mathbb{P}(1^{n+1},2)$, or a cubic hypersurface in $\mathbb{P}=\mathbb{P}^{n+1}$, or an intersection of two quadrics in $\mathbb{P}=\mathbb{P}^{n+2}$.

Proof.

Assertions (i) and (ii) follow from assertions (ii) and (iii) of Proposition 3.7, respectively, applied together with Proposition 1.5. If $n=2$, assertion (iii) follows from Lemma 2.6. If $n\geqslant 3$, assertion (iii) follows from Proposition 3.7(iv) and Proposition 1.5. ∎

Remark 3.9*.*

An alternative way to prove Corollary 3.8 (which in turn can be used to deduce Proposition 3.7) is by induction on dimension using the classification of smooth well formed Fano weighted complete intersections of low dimension (say, one provided by Lemma 2.6) together with [PST17, Theorem 1.2].

4. Automorphisms

In this section we prove Theorem 1.3.

Let $\mathbb{P}=\mathbb{P}(a_{0},\ldots,a_{N})$ be a weighted projective space. For any subvariety $X\subset\mathbb{P}$, we denote by $\mathrm{Aut}(\mathbb{P};X)$ the stabilizer of $X$ in $\mathrm{Aut}(\mathbb{P})$. We denote by $\mathrm{Aut}_{\mathbb{P}}(X)$ the image of $\mathrm{Aut}(\mathbb{P};X)$ under the restriction map to $\mathrm{Aut}(X)$. In other words, the group $\mathrm{Aut}_{\mathbb{P}}(X)$ consists of automorphisms of $X$ induced by automorphisms of $\mathbb{P}$.

We start with a general result that is well known to experts (see for instance [KPS18, Lemma 3.1.2]) and that was pointed out to us by A. Massarenti.

Lemma 4.1**.**

Let $X$ be a normal variety, let $A$ be a very ample Weil divisor on $X$, and let $[A]$ be the class of $A$ in $\mathrm{Cl}\,(X)$. Denote by $\mathrm{Aut}(X;[A])$ the stabilizer of $[A]$ in $\mathrm{Aut}(X)$. Then $\mathrm{Aut}(X;[A])$ is a linear algebraic group.

Corollary 4.2**.**

Let $X$ be a quasi-smooth well formed weighted complete intersection of dimension $n$. Suppose that either $n\geqslant 3$, or $K_{X}\neq 0$. Then $\mathrm{Aut}(X)$ is a linear algebraic group.

Proof.

Note that the divisor class $K_{X}$ is $\mathrm{Aut}(X)$-invariant. Moreover, if $K_{X}\neq 0$, then either $K_{X}$ or $-K_{X}$ is ample by Theorem 2.5. On the other hand, if $n\geqslant 3$, then $\mathrm{Cl}\,(X)\cong\mathbb{Z}$ by Lemma 2.4, so that an ample generator of $\mathrm{Cl}\,(X)$ is $\mathrm{Aut}(X)$-invariant. In both cases we see that $\mathrm{Aut}(X)$ preserves some ample (and thus also some very ample) divisor class on $X$. Hence $\mathrm{Aut}(X)$ is a linear algebraic group by Lemma 4.1. ∎

The following lemma will not be used in the proof of Theorem 1.3, but will allow us to prove its (weaker) analog that applies to a slightly wider class of smooth weighted complete intersections, see Corollary 4.7(ii) below.

Lemma 4.3**.**

Let $X$ be a subvariety of $\mathbb{P}$. Then $\mathrm{Aut}_{\mathbb{P}}(X)$ is a linear algebraic group.

Proof.

The group $\mathrm{Aut}(\mathbb{P})$ is obviously a linear algebraic group. The stabilizer $\mathrm{Aut}(\mathbb{P};X)$ of $X$ in $\mathrm{Aut}(\mathbb{P})$ and the kernel $\mathrm{Aut}(\mathbb{P};X)_{\mathrm{id}}$ of its action on $X$ are cut out in $\mathrm{Aut}(\mathbb{P})$ by algebraic equations, so that they are linear algebraic groups. The group $\mathrm{Aut}(\mathbb{P};X)_{\mathrm{id}}$ is a normal subgroup of $\mathrm{Aut}(\mathbb{P};X)$. Therefore, the group

[TABLE]

is a linear algebraic group as well, see [Bo69, Theorem 6.8]. ∎

Corollary 4.4**.**

Let $X$ be a smooth irreducible subvariety of $\mathbb{P}$. Suppose that $K_{X}$ is numerically effective. Then the group $\mathrm{Aut}_{\mathbb{P}}(X)$ is finite.

Proof.

By Lemma 4.3, the group $\mathrm{Aut}_{\mathbb{P}}(X)$ is a linear algebraic group. Therefore, if $\mathrm{Aut}_{\mathbb{P}}(X)$ is infinite, then it contains a subgroup isomorphic either to $\Bbbk^{\times}$ or to $\Bbbk^{+}$, which implies that $X$ is covered by rational curves. On the other hand, since $K_{X}$ is numerically effective, $X$ can’t be covered by rational curves, see [MiMo86, Theorem 1]. ∎

The main tool we use in the proof of Theorem 1.3 is the following result from [Fle81].

Theorem 4.5** (see [Fle81, Satz 8.11(c)]).**

Let $X$ be a smooth weighted complete intersection of dimension $n\geqslant 2$. Then

[TABLE]

for every integer $i\leqslant n-2$.

Remark 4.6*.*

Actually, the assertion of [Fle81, Satz 8.11(c)] gives more vanishing results and holds under the weaker assumption that $X$ is quasi-smooth. However, we do not want to go into details with the definition of the sheaf $\Omega_{X}^{1}$ here, and in any case we will need smoothness of $X$ on the next step.

Theorem 4.5 allows us to prove finiteness of various automorphism groups.

Corollary 4.7**.**

Let $X$ be a smooth well formed weighted complete intersection of dimension $n\geqslant 2$. Suppose that $i_{X}\leqslant n-2$. The following assertions hold.

• (i)

One has $H^{0}(X,T_{X})=0$.

• (ii)

The group $\mathrm{Aut}_{\mathbb{P}}(X)$ is finite.

• (iii)

If either $\dim X\geqslant 3$ or $K_{X}\neq 0$, then the group $\mathrm{Aut}(X)$ is finite.

Proof.

By Theorem 4.5 we have

[TABLE]

Recall that $\omega_{X}=\mathcal{O}_{X}(-i_{X})$ by Theorem 2.5. Thus assertion (i) follows from Serre duality. Assertion (ii) follows from assertion (i), because $\mathrm{Aut}_{\mathbb{P}}(X)$ is a linear algebraic group by Lemma 4.3. Similarly, assertion (iii) follows from assertion (i), because the automorphism group of any variety subject to the above assumptions is a linear algebraic group by Corollary 4.2. ∎

Recall that the smooth Fano threefold $V_{5}$ of Fano index $\dim V_{5}-1=2$ that is defined as an intersection of the Grassmannian $\mathrm{Gr}(2,5)\subset\mathbb{P}^{9}$ in its Plücker embedding with a linear section of codimension $3$ has infinite automorphism group $\mathrm{Aut}(V_{5})\cong\mathrm{PGL}_{2}(\Bbbk)$, see [Mu88, Proposition 4.4] or [CS15, Proposition 7.1.10]. The next lemma shows that such a situation is impossible for smooth weighted complete intersections.

Lemma 4.8**.**

Let $X\subset\mathbb{P}$ be a smooth well formed weighted complete intersection of dimension $n\geqslant 2$. Suppose that $i_{X}=n-1$. Then the group $\mathrm{Aut}(X)$ is finite.

Proof.

If $n=2$, the assertion follows from Remark 2.7 and the properties of automorphism groups of smooth del Pezzo surfaces, see for instance [Do12, Corollary 8.2.40]. Thus, we assume that $n\geqslant 3$ and use the classification provided by Proposition 3.7(iv). If $X$ is isomorphic to an intersection of two quadrics in $\mathbb{P}=\mathbb{P}^{n+2}$ or to a cubic hypersurface in $\mathbb{P}=\mathbb{P}^{n+1}$, then the assertion follows from Theorem 1.2 (in the latter case one can also use Theorem 1.1).

Now suppose that $X$ is isomorphic either to a hypersurface of degree $4$ in $\mathbb{P}=\mathbb{P}(1^{n+1},2)$, or to a hypersurface of degree $6$ in $\mathbb{P}=\mathbb{P}(1^{n},2,3)$. The argument in these cases is similar to that in the proof of [KPS18, Lemma 4.4.1]. Denote by $H$ the ample divisor such that $-K_{X}\sim(n-1)H$. Then there exists an $\mathrm{Aut}(X)$-equivariant double cover $\phi\colon X\to Y$, where in the former case $Y\cong\mathbb{P}^{n}$ and $\phi$ is given by the linear system $|H|$, while in the latter case $Y\cong\mathbb{P}(1^{n},2)$ and $\phi$ is given by the linear system $|2H|$. Let $H^{\prime}$ be the ample Weil divisor generating the group $\mathrm{Cl}\,(Y)\cong\mathbb{Z}$, and let $B\subset Y$ be the branch divisor of $\phi$. In the former case one has $B\sim 4H^{\prime}$, and in the latter case one has $B\sim 6H^{\prime}$. Note that in the latter case $H^{\prime}$ is not Cartier, but $2H^{\prime}$ is; note also that in this case $\phi$ is branched over the singular point of $Y$ as well. In both cases $B$ is smooth. Furthermore, it follows from adjunction formula that either $K_{B}$ is ample, or $K_{B}\sim 0$, or $B$ is a (smooth well formed) Fano weighted hypersurface of dimension $n-1\geqslant 3$ and Fano index $i_{B}\leqslant n-3$.

Since the double cover $\phi$ is $\mathrm{Aut}(X)$-equivariant, we see that the quotient of the group $\mathrm{Aut}(X)$ by its normal subgroup of order $2$ generated by the Galois involution of $\phi$ is isomorphic to a subgroup of the stabilizer $\mathrm{Aut}(Y;B)$ of $B$ in $\mathrm{Aut}(Y)$. Since $B$ is not contained in any divisor linearly equivalent to the very ample divisor $2H^{\prime}$, we conclude that $\mathrm{Aut}(Y;B)$ acts faithfully on $B$, see for instance [CPS19, Lemma 2.1]. Hence

[TABLE]

On the other hand, the group $\mathrm{Aut}_{Y}(B)$ is finite by Corollary 4.7(ii); alternatively, one can apply Corollaries 4.4 and 4.7(iii). This means that the group $\mathrm{Aut}(X)$ is finite as well. ∎

Now we prove our main results.

Proof of Theorem 1.3.

First suppose that $n=1$. We may assume that $K_{X}$ is ample. In this case the finiteness of $\mathrm{Aut}(X)$ is well-known, see for instance [Ha77, Exercise IV.5.2].

Now suppose that $n\geqslant 2$. If $i_{X}\leqslant n-2$, then the group $\mathrm{Aut}(X)$ is finite by Corollary 4.7(iii). If $i_{X}=n-1$, then the group $\mathrm{Aut}(X)$ is finite by Lemma 4.8. Finally, if $i_{X}\geqslant n$, then we know from Proposition 3.7 that $X$ is isomorphic either to $\mathbb{P}^{n}$ or to a quadric hypersurface in $\mathbb{P}^{n+1}$. ∎

Corollary 1.4 immediately follows from Theorem 1.3 and Proposition 1.5.