Automorphisms of weighted complete intersections
aa r X i v : . [ m a t h . AG ] J a n AUTOMORPHISMS OF WEIGHTED COMPLETE INTERSECTIONS
VICTOR PRZYJALKOWSKI AND CONSTANTIN SHRAMOV
Abstract.
We show that smooth well formed weighted complete intersections have finiteautomorphism groups, with several obvious exceptions. 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 thefollowing 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 ⊂ P N , N > , be a smoothhypersurface of degree d > . Suppose that ( N, d ) = (3 , . Then the group 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 dimen-sion at least in P N that is not contained in a hyperlplane. Suppose that X does notcoincide with P N , is not a quadric hypersurface in P N , and is not a K surface. Then thegroup Aut( X ) is finite. The goal of this paper is to generalize Theorems 1.1 and 1.2 to the case of smoothweighted complete intersections. We refer the reader to [Do82] and [IF00] (or to § Theorem 1.3.
Let X be a smooth well formed weighted complete intersection of dimen-sion n . Suppose that either n > , or K X = 0 . Then the group Aut( X ) is finite unless X is isomorphic either to P n or to a quadric hypersurface in P n +1 . Under a minor additional assumption (cf. Definition 2.2 below) one can make the assertionof Theorem 1.3 more precise.
Corollary 1.4.
Let X ⊂ P be a smooth well formed weighted complete intersection ofdimension n that is not an intersection with a linear cone. Suppose that either n > ,or K X = 0 . Then the group Aut( X ) is finite unless X = P ∼ = P n or X is a quadrichypersurface in P ∼ = P n +1 . Note that if X is not an intersection with a linear cone, then the assumption of The-orem 1.3 is equivalent to the requirement that X is not one of the weighted completeintersections listed in Table 2 below. Victor Przyjalkowski was partially supported by Laboratory of Mirror Symmetry NRU HSE, RF Govern-ment grant, ag. № e refer the reader to [HMX13], [KPS18, Theorem 1.1.2], and [CPS19, Theorem 1.2] forother 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 (seeProposition 3.7(iv) and Lemma 4.8).To deduce Corollary 1.4 from Theorem 1.3, we need the the following assertion thatis 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 in-tersection X ⊂ P = P ( a , . . . , a N ) of multidegree ( d , . . . , d k ) is normalized if the inequali-ties a . . . a N and d . . . d k hold. Proposition 1.5 (cf. [IF00, Lemma 18.3]) . Let X ⊂ P ( a , . . . , a N ) and X ′ ⊂ P ( a ′ , . . . , a ′ N ′ ) be normalized quasi-smooth well formed weighted complete intersections of multidegrees ( d , . . . , d k ) and ( d ′ , . . . , d ′ k ′ ) , respectively, such that X and X ′ are not intersections withlinear cones. Suppose that X ∼ = X ′ and dim X > . Then N = N ′ , k = k ′ , a i = a ′ i for every i N , and d j = d ′ j for every j k . When the first draft of this paper was completed, A. Massarenti informed us that a resultessentially similar to Theorem 1.3 was proved earlier in [ACM18, Proposition 5.7]. Notehowever that in [ACM18, §
5] the authors work with smooth weighted complete intersectionssubject to certain strong additional assumptions (see [ACM18, Assumptions 5.2] for details).Throughout the paper we work over an algebraically closed field k of characteristic zero.The plan of our paper is as follows. In § § § Preliminaries
In this section we recall the basic properties of weighted complete intersections. Werefer the reader to [Do82] and [IF00] for more details. Some properties of smooth weightedcomplete intersections can be also found in the earlier paper [Mo75].Let a , . . . , a N be positive integers. Consider the graded algebra k [ x , . . . , x N ], where thegrading is defined by assigning the weights a i to the variables x i . Put P = P ( a , . . . , a N ) = Proj k [ x , . . . , x N ] . The weighted projective space P is said to be well formed if the greatest common divisorof any N of the weights a i is 1. Every weighted projective space is isomorphic to a wellformed one, see [Do82, 1.3.1]. A subvariety X ⊂ P is said to be well formed if P is wellformed and codim X ( X ∩ Sing P ) > , where the dimension of the empty set is defined to be − X ⊂ P of codimension k is a weighted complete intersection ofmultidegree ( d , . . . , d k ) if its weighted homogeneous ideal in k [ x , . . . , x N ] is generated by aregular sequence of k homogeneous elements of degrees d , . . . , d k . This is equivalent to the equirement that the codimension of (every irreducible component of) the variety X equalsthe (minimal) number of generators of the weighted homogeneous ideal of X , cf. [Ha77,Theorem II.8.21A(c)]. Note that P can be thought of as a weighted complete intersectionof codimension 0 in itself; this gives us a smooth Fano variety if and only if P ∼ = P N . Definition 2.1 (see [IF00, Definition 6.3]) . Let p : A N +1 \{ } → P be the natural projectionto the weighted projective space. A subvariety X ⊂ P is called quasi-smooth if p − ( X ) issmooth.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 certainextent analogous to complete intersections in a usual projective space that are contained ina hyperplane. Definition 2.2 (cf. [IF00, Definition 6.5]) . A weighted complete intersection X ⊂ P is saidto be an intersection with a linear cone if one has d j = a i for some i and j . Remark . A general quasi-smooth well formed weighted complete intersection is isomor-phic to a quasi-smooth well formed weighted complete intersection that is not an intersec-tion with a linear cone, cf. [PSh16, Remark 5.2]. Note however that this does not holdwithout the generality assumption. For instance, a general weighted complete intersec-tion of bidegree (2 ,
4) in P (1 n +2 ,
2) is isomorphic to a quartic hypersurface in P n +1 , whilecertain 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 ⊂ P , we denote by O X (1) the restriction of the sheaf O P (1) to X ,see [Do82, 1.4.1]. Note that the sheaf O P (1) may be not invertible. However, if X is wellformed, then O X (1) is a well-defined divisorial sheaf on X . Furthermore, if X is well formedand smooth, then 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 .Then the class of the divisorial sheaf O X (1) generates the group Cl ( X ) of classes of Weildivisors on X . In particular, under the additional assumption that X is smooth, the classof the line bundle O X (1) generates the group Pic ( X ) . One can describe the canonical class of a weighted complete intersection. For a weightedcomplete intersection X of multidegree ( d , . . . , d k ) in P , define i X = X a j − X d i . Let ω 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 wellformed weighted complete intersection. Then ω X = O X ( − i X ) . Using the bounds on numerical invariants of smooth weighted complete intersections foundin [CCC11, Theorem 1.3], [PSh16, Theorem 1.1], and [PST17, Corollary 5.3(i)], one caneasily obtain the classically known lists of all smooth Fano weighted complete intersectionsof small dimensions. Namely, we have the following. emma 2.6. Let X be a smooth well formed Fano weighted complete intersection of dimen-sion at most in P that is not an intersection with a linear cone. Then X is one of thevarieties listed in Table 1. No. P Degreesdimension 11.1 P P ∅ dimension 22.1 P (1 , ,
3) 62.2 P (1 ,
2) 42.3 P P , P P ∅ Table 1: Fano weighted complete intersections in dimen-sions 1 and 2
Remark . Let X be a smooth well formed Fano weighted complete intersection of dimen-sion 2. If we do not assume that X is not an intersection with a linear cone, we cannotuse the classification provided by Lemma 2.6. However, Lemma 2.6 applied together withRemark 2.3 shows that if i X = 1, then X is a del Pezzo surface of (anticanonical) degree atmost 4.Recall that the Fano index of a Fano variety X is defined as the maximal integer m suchthat the canonical class K X is divisible by m in the Picard group of X . Theorem 2.5 andLemmas 2.4 and 2.6 imply the following. Corollary 2.8.
Let X be a smooth Fano well formed weighted complete intersection ofdimension at least . 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 ofsmooth weighted complete intersections of a given multidegree in a given weighted projectivespace. 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 classificationprovided in Lemma 2.6.If dim X >
3, then we apply Theorem 2.5 together with Lemma 2.4. (cid:3)
Note that the assertion of Corollary 2.8 fails in dimension 1: if X is a conic in P ,then i X = 1, while the Fano index of X equals 2. emma 2.9. Let X ⊂ P ( a , . . . , a N ) be a smooth well formed weighted complete intersectionof multidegree ( d , . . . , d k ) . Then a general weighted complete intersection X ′ of multidegree ( d , . . . , d k ) in P (1 , a , . . . , a N ) is smooth and well formed, and i X ′ = i X + 1 .Proof. Straightforward. (cid:3)
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 Fanoweighted 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 weightedcomplete intersections of dimension up to 2 with trivial canonical class.
Lemma 2.10.
Let X be a smooth well formed weighted complete intersection of dimensionat most in 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 2. No. P Degreesdimension 11.1 P (1 , ,
3) 61.2 P (1 , ,
2) 41.3 P P , P (1 ,
3) 62.2 P P , P , , Remark . Note that each of the four families of elliptic curves listed in Table 2 in factcontains all elliptic curves up to isomorphism. This is not the case for K Uniqueness of embeddings
In this section we prove Proposition 1.5. Let us start with a couple of facts that are wellknown and can be proved similarly to their analogs for complete intersections in the usualprojective space. However, we provide their proofs for the reader’s convenience.The proof of the following was suggested to us by A. Kuznetsov. emma 3.1. Let X ⊂ P = P ( a , . . . , a N ) be a subvariety. Let C ∗ X be a complement of theaffine cone over X to its vertex, and let p : C ∗ X → X be the projection. Then one has p ∗ ( O C ∗ X ) = M m ∈ Z O X ( m ) . Proof.
Denote the polynomial ring k [ x , . . . , x N ] by R and let U = (cid:0) Spec R (cid:1) \ { } ∼ = A N +1 \ { } . Let Y = Spec P M m > O P ( m ) ! be the relative spectrum. Since R ∼ = Γ (cid:0)L m > O P ( m ) (cid:1) , one obtains the map Y → Spec R ∼ = A N +1 which is a weighted blow up of the origin (with weights a , . . . , a N ). Cutting out the originone gets the isomorphism U ∼ = Spec P M m ∈ Z O P ( m ) ! . Consider the fibered product C ∗ X ∼ = X × P U. Taking into account that O P ( m ) | X = O X ( m ) by definition, we obtain an isomorphism C ∗ X ∼ = Spec X M m ∈ Z O X ( m ) ! , and the assertion of the lemma follows. (cid:3) For a subvariety X ⊂ P ( a , . . . , a N ), we define the Poincar´e series P X ( t ) = X m > h ( X, O X ( m )) t m . 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, Theo-rem 3.2.4(iii)]) . Let X ⊂ P ( a , . . . , a N ) be a weighted complete intersection of multideg-ree ( d , . . . , d k ) . The following assertions hold. (i) One has P X ( t ) = Q kj =1 (1 − t d j ) Q Ni =0 (1 − t a i ) . (ii) One has H i ( X, O X ( m )) = 0 for all m ∈ Z and all < i < dim X .Proof. Denote the graded polynomial ring k [ x , . . . , x N ] by R , denote the weighted homo-geneous ideal ( f , . . . , f k ) that defines X by I , and put S = R/I , so that X ∼ = Proj S . Fromregularity of the sequence f , . . . , f k it easily follows that(3.1) X m > dim( S m ) t m = Q kj =1 (1 − t d j ) Q Ni =0 (1 − t a i ) , here S m is the m -th graded component of S .Denote the affine cone Spec S ⊂ Spec R ∼ = A N +1 over X by C X . By construction, one has an isomorphism of graded algebras(3.2) S ∼ = H ( C X , O C X ) . Following Lemma 3.1 denote the complement of C X to its vertex P by C ∗ X . Consider thelocal cohomology groups H • P ( C X , O C X ), see, for instance, [Ha67, p. 2, Definition]. We havethe exact sequence . . . → H iP ( C X , O C X ) → H i ( C X , O C X ) → H i (cid:0) C ∗ X , O C ∗ X (cid:1) → H i +1 P ( C X , O C X ) → . . . , see [Ha67, Corollary 1.9]. Since C X is a complete intersection in the affine space, it isCohen–Macaulay. Therefore, by [Ha67, Proposition 3.7] and [Ha67, Theorem 3.8] one has H iP ( C X , O C X ) = 0for all i < dim C X = dim X + 1. Hence, we obtain an isomorphism(3.3) H i ( C X , O C X ) ∼ = H i (cid:0) C ∗ X , O C ∗ X (cid:1) for all i < dim X .Finally, denote the natural projection C ∗ X → X by p . Then(3.4) H i ( C ∗ X , O C ∗ X ) ∼ = H i ( X, p ∗ O C ∗ X )for all i . On the other hand, by Lemma 3.1 we have(3.5) p ∗ ( O C ∗ X ) = M m ∈ Z O X ( m ) . For i = 0, we combine the isomorphisms (3.2), (3.3), (3.4), and (3.5) to obtain an isomor-phism of graded algebras S ∼ = M m > H ( X, O X ( m )) . 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 anisomorphism H i ( C X , O C X ) ∼ = M m ∈ Z H i ( X, O X ( m )) . Since C X is an affine variety, we have H i ( C X , O C X ) = 0 for all i >
0, see for in-stance [Ha77, Theorem III.3.5]. This proves assertion (ii). (cid:3)
Proposition 3.2(i) implies the following property that can be considered as an analog oflinear normality for usual complete intersections.
Corollary 3.3.
Let X ⊂ P be a quasi-smooth well formed weighted complete intersection.Then the restriction map H (cid:0) P , O P ( m ) (cid:1) → H (cid:0) X, O X ( m ) (cid:1) is surjective for every m ∈ Z .Proof. The dimension of the image of the restriction map is computed by the coefficientin the Poincar´e series (3.1). On the other hand, by Proposition 3.2(i) this coefficient alsoequals the dimension of H ( X, O X ( m )). (cid:3) o proceed, we will need an elementary observation. Lemma 3.4 (see [IF00, Lemma 18.3]) . Let N and N ′ be positive integers, and k and k ′ benon-negative integers. Let a . . . a N , a ′ . . . a ′ N ′ , d . . . d k , and d ′ . . . d ′ k ′ be positive integers. Suppose that (3.6) Q kj =1 (1 − t d j ) Q Ni =0 (1 − t a i ) = Q k ′ j ′ =1 (1 − t d ′ j ′ ) Q N ′ i ′ =0 (1 − t a ′ i ′ ) as rational functions in the variable t . Suppose that a i = d j for all i and j , and a ′ i ′ = d ′ j ′ for all i ′ and j ′ . Then N = N ′ , k = k ′ , a i = a ′ i for every i N , and d j = d ′ j forevery j k .Proof. Note that numerators and denominators of the rational functions in the left and theright hand sides of (3.6) may have common divisors (for instance, if some d j is divisible bysome a i , or another way around). To prove the assertion, we will keep track of the numbersthat 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 interchang-ing the collections { a i } , { a ′ j } with { d s } , { d ′ r } , respectively. Thus we may assume that d k isthe maximal number among a N , a ′ N ′ , d k , and d ′ k ′ . By assumption we know that a N < d k .Let ζ be a primitive d k -th root of unity. Then ζ is a root of the numerator of the left handside of (3.6) but not the root of its denominator. Hence ζ is a root of the numerator ν ( t )of the right hand side of (3.6) as well. Since d ′ j ′ d k for all j ′ , we see that ν ( t ) is divisibleby 1 − t d k . Cancelling the factor 1 − t d k from (3.6), we complete the proof of the lemma byinduction. (cid:3) Now we prove the main result of this section.
Proof of Proposition 1.5.
By Lemma 2.4, the group Cl ( X ) is generated by the class ofthe line bundle O X (1), while the group Cl ( X ′ ) is generated by the class of the line bun-dle O X ′ (1). Therefore, we see from Proposition 3.2(i) that Q kj =1 (1 − t d j ) Q Ni =0 (1 − t a i ) = P X ( t ) = P X ′ ( t ) = Q k ′ j ′ =1 (1 − t d ′ j ′ ) Q N ′ i ′ =0 (1 − t a ′ i ′ ) . Since neither X nor X ′ is an intersection with a linear cone, the required assertion followsfrom Lemma 3.4. (cid:3) Remark . The assertion of Proposition 1.5 also holds for smooth Fano weighted completeintersections 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 P is isomorphicto P (which can be considered as a complete intersection of codimension 0 in itself). Remark . We point out that the assumption that X is Fano is essential for thevalidity of Proposition 1.5 in dimension 2. For instance, there exist smooth quarticsin P that also have a structure of a double cover of P branched in a sextic curve, seee.g. [MaMo63, Proof of Theorem 4]. Similarly, by Remark 2.11 the assertion of Proposi-tion 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 ′ 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 P (1 , , which can be considered as a quasi-smooth well formed weighted complete intersectionof codimension 0 in itself) can be embedded as a quadratic cone into P (which is notquasi-smooth).In the proof of Theorem 1.3, we will need a classification of weighted complete intersectionsof large Fano index. Proposition 3.7 (cf. [PSh18, Theorem 2.7]) . Let X ⊂ P be a smooth well formed Fanoweighted complete intersection of dimension n > . Then (i) one has i X n + 1 ; (ii) if i X = n + 1 , then X ∼ = P n ; (iii) if i X = n , then X is isomorphic to a quadric in P n +1 ; (iv) if i X = n − and n > , then X is isomorphic to a hypersurface of degree in P = P (1 n , , , or to a hypersurface of degree in P = P (1 n +1 , , or to a cubichypersurface in P = P n +1 , or to an intersection of two quadrics in P = 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 n + 1; if i X = n + 1, then X is iso-morphic to P n ; and if i X = n , then X is isomorphic to a quadric in P n +1 . This provesassertions (i), (ii), and (iii).Now suppose that i X = n − n >
3. Note that Pic ( X ) ∼ = Z by Lemma 2.4. Thus itfollows from the classification of smooth Fano varieties of Fano index n − X is isomorphic either to one of the weighted completeintersections listed in assertion (iv), or to a linear section of the Grassmannian Gr(2 , X cannot be isomorphic to alinear section of the Grassmannian Gr(2 , X is isomorphic to such a variety.Then n
6. If n = 3, then the Fano index of X equals 2 and its anticanonical degree isequal to 40. If n = 4, then the Fano index of X equals 3 and its anticanonical degree isequal to 405. In both cases we see from Remark 2.3 that there exists a smooth weightedcomplete intersection X with the same n and i X that is not an intersection with a linearcone. This is impossible by a classification of smooth Fano weighted complete intersections ofdimensions 3 and 4, see [PSh18, Table 2] (cf. [IP99, § n
6. Recall that H (cid:0) Gr(2 , , Z (cid:1) ∼ = Z . Thus, Lefschetz hyperplane section theorem implies that H ( X, Z ) ∼ = Z . On the other hand, since X is a weighted complete intersection of dimension greater than 4,one has H ( X, Z ) ∼ = Z by the Lefschetz-type theorem for complete intersections in toricvarieties, see [Ma99, Proposition 1.4]. The obtained contradiction completes the proof ofassertion (iv). (cid:3) Proposition 1.5 allows us to prove more precise classification results concerning Fanoweighted complete intersections (which we will not use directly in our further proofs).
Corollary 3.8.
Let X ⊂ P be a smooth well formed Fano weighted complete intersection ofdimension n > that is not an intersection with a linear cone. Then (i) if i X = n + 1 , then X = P = P n ; ii) if i X = n , then X is a quadric in P = P n +1 ; (iii) if i X = n − , then X is either a hypersurface of degree in P = P (1 n , , , or ahypersurface of degree in P = P (1 n +1 , , or a cubic hypersurface in P = P n +1 , oran intersection of two quadrics in P = P n +2 .Proof. Assertions (i) and (ii) follow from assertions (ii) and (iii) of Proposition 3.7, respec-tively, applied together with Proposition 1.5. If n = 2, assertion (iii) follows from Lemma 2.6.If n >
3, assertion (iii) follows from Proposition 3.7(iv) and Proposition 1.5. (cid:3)
Remark . An alternative way to prove Corollary 3.8 (which in turn can be used to deduceProposition 3.7) is by induction on dimension using the classification of smooth well formedFano 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 P = P ( a , . . . , a N ) be a weighted projective space. For any subvariety X ⊂ P , wedenote by Aut( P ; X ) the stabilizer of X in Aut( P ). We denote by Aut P ( X ) the imageof Aut( P ; X ) under the restriction map to Aut( X ). In other words, the group Aut P ( X )consists of automorphisms of X induced by automorphisms of P .We start with a general result that is well known to experts (see for in-stance [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 , andlet [ A ] be the class of A in Cl ( X ) . Denote by Aut( X ; [ A ]) the stabilizer of [ A ] in Aut( X ) .Then Aut( X ; [ A ]) is a linear algebraic group. Corollary 4.2.
Let X be a quasi-smooth well formed weighted complete intersection ofdimension n . Suppose that either n > , or K X = 0 . Then Aut( X ) is a linear algebraicgroup.Proof. Note that the divisor class K X is Aut( X )-invariant. Moreover, if K X = 0, theneither K X or − K X is ample by Theorem 2.5. On the other hand, if n >
3, then Cl ( X ) ∼ = Z by Lemma 2.4, so that an ample generator of Cl ( X ) is Aut( X )-invariant. In both cases wesee that Aut( X ) preserves some ample (and thus also some very ample) divisor class on X .Hence Aut( X ) is a linear algebraic group by Lemma 4.1. (cid:3) The following lemma will not be used in the proof of Theorem 1.3, but will allow us toprove its (weaker) analog that applies to a slightly wider class of smooth weighted completeintersections, see Corollary 4.7(ii) below.
Lemma 4.3.
Let X be a subvariety of P . Then Aut P ( X ) is a linear algebraic group.Proof. The group Aut( P ) is obviously a linear algebraic group. The stabilizer Aut( P ; X )of X in Aut( P ) and the kernel Aut( P ; X ) id of its action on X are cut out in Aut( P ) byalgebraic equations, so that they are linear algebraic groups. The group Aut( P ; X ) id is anormal subgroup of Aut( P ; X ). Therefore, the groupAut P ( X ) ∼ = Aut( P ; X ) / Aut( P ; X ) id is a linear algebraic group as well, see [Bo69, Theorem 6.8]. (cid:3) orollary 4.4. Let X be a smooth irreducible subvariety of P . Suppose that K X is numer-ically effective. Then the group Aut P ( X ) is finite.Proof. By Lemma 4.3, the group Aut P ( X ) is a linear algebraic group. Therefore, if Aut P ( X )is infinite, then it contains a subgroup isomorphic either to k × or to k + , which implies that X is covered by rational curves. On the other hand, since K X is numerically effective, X can’tbe covered by rational curves, see [MiMo86, Theorem 1]. (cid:3) 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 intersectionof dimension n > . Then H n (cid:0) X, Ω X ⊗ O X ( − i ) (cid:1) = 0 for every integer i n − .Remark . Actually, the assertion of [Fle81, Satz 8.11(c)] gives more vanishing resultsand holds under the weaker assumption that X is quasi-smooth. However, we do not wantto go into details with the definition of the sheaf Ω X here, and in any case we will needsmoothness 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 dimen-sion n > . Suppose that i X n − . The following assertions hold. (i) One has H ( X, T X ) = 0 . (ii) The group
Aut P ( X ) is finite. (iii) If either dim X > or K X = 0 , then the group Aut( X ) is finite.Proof. By Theorem 4.5 we have H n (cid:0) X, Ω X ⊗ O X ( − i X ) (cid:1) = 0 . Recall that ω X = O X ( − i X ) by Theorem 2.5. Thus assertion (i) follows from Serre dual-ity. Assertion (ii) follows from assertion (i), because Aut P ( X ) is a linear algebraic groupby Lemma 4.3. Similarly, assertion (iii) follows from assertion (i), because the automor-phism group of any variety subject to the above assumptions is a linear algebraic group byCorollary 4.2. (cid:3) Recall that the smooth Fano threefold V of Fano index dim V − , ⊂ P in its Pl¨ucker embedding witha linear section of codimension 3 has infinite automorphism group Aut( V ) ∼ = PGL ( k ),see [Mu88, Proposition 4.4] or [CS15, Proposition 7.1.10]. The next lemma shows that sucha situation is impossible for smooth weighted complete intersections. Lemma 4.8.
Let X ⊂ P be a smooth well formed weighted complete intersection of dimen-sion n > . Suppose that i X = n − . Then the group Aut( X ) is finite.Proof. If n = 2, the assertion follows from Remark 2.7 and the properties of automorphismgroups of smooth del Pezzo surfaces, see for instance [Do12, Corollary 8.2.40]. Thus, weassume that n > X is isomor-phic to an intersection of two quadrics in P = P n +2 or to a cubic hypersurface in P = P n +1 ,then the assertion follows from Theorem 1.2 (in the latter case one can also use Theorem 1.1). ow suppose that X is isomorphic either to a hypersurface of degree 4 in P = P (1 n +1 , P = P (1 n , , H the ample divisor suchthat − K X ∼ ( n − H . Then there exists an Aut( X )-equivariant double cover φ : X → Y ,where in the former case Y ∼ = P n and φ is given by the linear system | H | , while in thelatter case Y ∼ = P (1 n ,
2) and φ is given by the linear system | H | . Let H ′ be the ample Weildivisor generating the group Cl ( Y ) ∼ = Z , and let B ⊂ Y be the branch divisor of φ . Inthe former case one has B ∼ H ′ , and in the latter case one has B ∼ H ′ . Note that inthe latter case H ′ is not Cartier, but 2 H ′ is; note also that in this case φ is branched overthe singular point of Y as well. In both cases B is smooth. Furthermore, it follows fromadjunction formula that either K B is ample, or K B ∼
0, or B is a (smooth well formed)Fano weighted hypersurface of dimension n − > i B n − φ is Aut( X )-equivariant, we see that the quotient of thegroup Aut( X ) by its normal subgroup of order 2 generated by the Galois involution of φ is isomorphic to a subgroup of the stabilizer Aut( Y ; B ) of B in Aut( Y ). Since B is notcontained in any divisor linearly equivalent to the very ample divisor 2 H ′ , we concludethat Aut( Y ; B ) acts faithfully on B , see for instance [CPS19, Lemma 2.1]. HenceAut( Y ; B ) ∼ = Aut Y ( B ) . On the other hand, the group Aut Y ( B ) is finite by Corollary 4.7(ii); alternatively, one canapply Corollaries 4.4 and 4.7(iii). This means that the group Aut( X ) is finite as well. (cid:3) 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 thiscase the finiteness of Aut( X ) is well-known, see for instance [Ha77, Exercise IV.5.2].Now suppose that n >
2. If i X n −
2, then the group Aut( X ) is finite by Corol-lary 4.7(iii). If i X = n −
1, then the group Aut( X ) is finite by Lemma 4.8. Finally, if i X > n ,then we know from Proposition 3.7 that X is isomorphic either to P n or to a quadric hy-persurface in P n +1 . (cid:3) Corollary 1.4 immediately follows from Theorem 1.3 and Proposition 1.5.
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Victor Przyjalkowski
Steklov Mathematical Institute of RAS, 8 Gubkina street, Moscow 119991, Russia.HSE University, Russian Federation, Laboratory of Mirror Symmetry, NRU HSE, 6 Usacheva str., Moscow,Russia, 119048. [email protected], [email protected]
Constantin Shramov
Steklov Mathematical Institute of RAS, 8 Gubkina street, Moscow 119991, Russia.HSE University, Russian Federation, Laboratory of Algebraic Geometry, 6 Usacheva str., Moscow, 119048,Russia. [email protected]@gmail.com