On abelian automorphism groups of hypersurfaces

Zheng, Zhiwei

doi:10.1007/s11856-021-2275-1

Cited by 11 publications

(9 citation statements)

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“…Since this is also a finite cyclic group we have that it is generated by an element s α = s α . Now since r d ≥ 4 by [Zhe20] Lemma 2.3 we have that there exist a smooth curve with an element of order (r d − 1) n in its automorphism group with presentation…”

Section: ) the Natural Map Pic([mentioning

confidence: 96%

Tautological families of cyclic covers of projective spaces

Kundu¹,

Mukherjee²,

Raychaudhury³

2021

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In this article, we study the existence of tautological families on a Zariski open set of the coarse moduli space parametrizing certain Galois covers over projective spaces. More specifically, let (1) H n.r.d (resp. M n,r,d ) be the stack (resp. coarse moduli) parametrizing smooth simple cyclic covers of degree r over the projective space P n branched along a divisor of degree r d ≥ 4, and (2) d 2 ) be the stack (resp. coarse moduli) of smooth cyclic triple covers over P 1 with 2d 1 − d 2 ≥ 4 and 2d 2 − d 1 ≥ 4. In the former case, we show that such a family exists if and only if gcd(r d,n +1) | d while in the latter case, we show that it always exists. We further show that even when such a family exists, often it cannot be extended to the open locus of objects without extra automorphisms. The existence of tautological families on a Zariski open set of its coarse moduli can be interpreted in terms of rationality of the stack if the coarse moduli space is rational. Combining our results with known results on the rationality of the coarse moduli of points on P 1 and the coarse moduli of plane curves (see [BG10], [BGK09]), we determine the rationality of H 1,r,d (resp. H 2,r,d ) for r d ≥ 4 (resp. r d ≥ 49). On the other hand H 1,3,d 1 ,d 2 is unirational, and we show that its coarse moduli M 1,3,d 1 ,d 2 is unirational and fibred over a rational base by homogeneous varieties which are rational if char(k) = 0. Our study is motivated by [GV09] and [GV08].

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Section: ) the Natural Map Pic([mentioning

confidence: 96%

Tautological families of cyclic covers of projective spaces

Kundu¹,

Mukherjee²,

Raychaudhury³

2021

Preprint

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“…By Definition 2.4, a set S of monomials is called simple, if S contains the monomials of a simple cubic polynomial in x 1 , • • • , x N . The terminology simple used here is slightly different from that is used in [Zhe21]. In [Zhe21] we did not consider polynomials of type Y .…”

Section: Cubic Fourfoldsmentioning

confidence: 99%

“…The terminology simple used here is slightly different from that is used in [Zhe21]. In [Zhe21] we did not consider polynomials of type Y . The following lemma is clear.…”

Section: Cubic Fourfoldsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In [Zhe21], the second author studied the abelian automorphism groups of smooth hypersurfaces. The method we use in this paper to prove Theorem 4.2 and 5.1 is an extension and refinement of the method in [Zhe21]. Our approach is based on the fact that a linear action of an abelian group on C N can be diagonalized.…”

Section: Introductionmentioning

confidence: 99%

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Abelian Automorphism Groups of Quartic Surfaces and Cubic Fourfolds

Peng,

Zheng

2021

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We classify (Theorem 4.2) abelian groups that admit a liftable action on a smooth cubic fourfold. Parallel result (Theorem 5.1) is obtained for quartic surfaces.the smoothness of F , we know that for any x i , there exists x j such that x d−1 i x j ∈ S(F ) (see Lemma 2.5). For a set S of monomials in x 1 , • • • , x N of degree d, we define a directed graph with [N] = {1, • • • , N} the set of vertices, and E S = {(i, j)|x d−1 i x j ∈ S} the set of arrows (from i to j). Therefore, if a homogeneous polynomial F is smooth, every vertex of the graph E S(F ) has at least one outlet. With a more detailed analysis of the structure of E S I , we will eventually prove Theorem 4.2 and 5.1.Stucture of the paper: In §2 we introduce some terminologies and preliminary results that will be used throughout the paper. In §3 we specialize to the study of automorphisms of cubic hypersurfaces. In §4 we concentrate on the study of cubic fourfolds and prove Theorem 4.2. Finally in §5 we consider the case of quartic surfaces.Acknowledgement: This work is the outcome of the first author's PCP project in 2020. We thank the staff of PCP for their support. The second author thanks MPIM for its support and excellent academic atmosphere. Preliminary ResultsGiven integers d 3, N 3. In this section, we fix some terminologies about (N − 2)folds of degree d and their automorphism groups. We denote by V N a complex vector space of dimension N, and denote by P N −1 the projectivization of V N . Let GL(N) be the group of linear automorphisms of V N , and let PGL(N) be the group of linear automorphisms of P N −1 . There is a natural group homomorphism GL(N) → PGL(N).For a global section F ∈ H 0 (P N −1 , O(d)), we denote by V (F ) ⊂ P N −1 the hypersurface of degree d defined by the zero locus of F . Let Lin(V (F )) be the group of linear automorphisms of P N −1 preserving V (F ). By Matsumura-Monsky [MM63], the group Lin(V (F )) is finite when V (F ) is smooth. Moreover, if V (F ) is smooth, N ≥ 4 and (d, N) = (4, 4), then Lin(V (F )) is equal to the group of regular automorphisms of V (F ) as a complex variety.Definition 2.1. Let V (F ) ⊂ P N −1 be an (N − 2)-fold of degree d. Take a subgroup G Lin(V (F )) PGL(N). The action of G on V (F ) is called liftable, if the group embedding G ֒→ PGL(N) factors through GL(N) → PGL(N) (Namely, there exists a group homomorphism G → GL(N), such that the composition of G → GL(N) → PGL(N) equals to the given one). Here, the group embedding G ֒→ GL(N) is called a lifting of the action. Given a coordinate system (x 1 , • • • , x N ), we call a linear automorphism of V N diagonal with respect to this coordinate system, if it sends2.1. Poset P. Now we take a finite abelian group G. Suppose we have a liftable action of G on a smooth (N − 2)-fold V (F ) of degree d. We choose a lifting ϕ 1 : G ֒→ GL(N). Since G is abelian, by standard linear algebra, we can choose a coordinate system (x 1 , • • • , x N ) of V N , such that all elements in G is diagonal with respect to (x 1 , • • • , x N ). Therefore, we have N charac...

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