Finite Group Of Automorphisms Research Articles

The field of moduli of varieties with a structure

AbstractIf X is a variety with an additional structure $$\xi $$ ξ , such as a marked point, a divisor, a polarization, a group structure and so forth, then it is possible to study whether the pair $$(X,\xi )$$ ( X , ξ ) is defined over the field of moduli. There exists a precise definition of “algebraic structures” which covers essentially all of the obvious concrete examples. We prove several formal results about algebraic structures. There are immediate applications to the study of fields of moduli of curves and finite sets in $$\mathbb {P}^{2}$$ P 2 , but the results are completely general. Fix G a finite group of automorphisms of X, a G-structure is an algebraic structure with automorphism group equal to G. First, we prove that G-structures on X are in a 1 : 1 correspondence with twisted forms of $$X/G\dashrightarrow \mathscr {B}G$$ X / G ⤏ B G . Secondly we show that, under some assumptions, every algebraic structure on X is equivalent to the structure given by some 0-cycle. Third, we give a cohomological criterion for checking the existence of G-structures not defined over the field of moduli. Fourth, we identify geometric conditions about the action of G on X which ensure that every G-structure is defined over the field of moduli.

Coble surfaces in characteristic two

We study Coble surfaces in characteristic 2, in particular, singularities of their canonical coverings. As an application we classify Coble surfaces with finite automorphism group in characteristic 2. There are exactly 9 types of such surfaces.

On K3 surfaces of Picard rank 14

AbstractWe study complex algebraic K3 surfaces with finite automorphism groups and polarized by rank 14, 2‐elementary lattices. Three such lattices exist, namely, , , and . As part of our study, we provide birational models for these surfaces as quartic projective hypersurfaces and describe the associated coarse moduli spaces in terms of suitable modular invariants. Additionally, we explore the connection between these families and dual K3 families related via the Nikulin construction.

Associative algebras of [formula omitted]-graded vertex operator superalgebras and their applications

Let V be a Z-graded vertex operator superalgebra and G a finite automorphism group of V of order T. For g∈G and n∈1TZ+, a series of associative algebras Ag,n(V) are constructed to study the twisted representations of V. In addition, we introduce a G-twisted associative algebra AG,n(V) such that Ag,n(V) is a quotient algebra of AG,n(V) for any g∈G. As an application of the theory of AG,n(V), we obtain a duality theorem of Schur-Weyl type. In particular, for any g∈G, every irreducible g-twisted V-module is a completely reducible VG-module.

Centralisers of finite groups in locally finite simple groups

We answer in the negative a question of Hartley about representations of finite groups, by constructing examples of finite simple groups with arbitrarily large representations whose endomorphism ring consists of just the scalars. We show as a consequence that there are finite simple groups of automorphisms of the locally finite simple group with trivial centralizer. The smallest of our examples is with .

Mori dream K3 surfaces of Picard number four: projective models and Cox rings

In this paper we study the geometry of the 14 families of K3 surfaces of Picard number four with finite automorphism group, whose Neron—Severi lattices have been classified by È. B. Vinberg. We provide projective models, we identify the degrees of a generating set of the Cox ring and in some cases we prove the unirationality of the associated moduli space.

An atlas of K3 surfaces with finite automorphism group

We study the geometry of the K3 surfaces $X$ with a finite number automorphisms and Picard number $\geq 3$. We describe these surfaces classified by Nikulin and Vinberg as double covers of simpler surfaces or embedded in a projective space. We study moreover the configurations of their finite set of $(-2)$-curves.

A Computational View on the Non-degeneracy Invariant for Enriques Surfaces

For an Enriques surface S, the non-degeneracy invariant retains information on the elliptic fibrations of S and its polarizations. In the current paper, we introduce a combinatorial version of the non-degeneracy invariant which depends on S together with a configuration of smooth rational curves, and gives a lower bound for . We provide a SageMath code that computes this combinatorial invariant and we apply it in several examples. First we identify a new family of nodal Enriques surfaces satisfying which are not general and with infinite automorphism group. We obtain lower bounds on for the Enriques surfaces with eight disjoint smooth rational curves studied by Mendes Lopes–Pardini. Finally, we recover Dolgachev and Kondō’s computation of the non-degeneracy invariant of the Enriques surfaces with finite automorphism group and provide additional information on the geometry of their elliptic fibrations.

Products of elliptic curves and abelian surfaces by finite groups of automorphisms

Products of elliptic curves and abelian surfaces by finite groups of automorphisms

Auslander’s theorem for dihedral actions on preprojective algebras of type A

AbstractGiven an algebra R and G a finite group of automorphisms of R, there is a natural map $\eta _{R,G}:R\#G \to \mathrm {End}_{R^G} R$ , called the Auslander map. A theorem of Auslander shows that $\eta _{R,G}$ is an isomorphism when $R=\mathbb {C}[V]$ and G is a finite group acting linearly and without reflections on the finite-dimensional vector space V. The work of Mori–Ueyama and Bao–He–Zhang has encouraged the study of this theorem in the context of Artin–Schelter regular algebras. We initiate a study of Auslander’s result in the setting of nonconnected graded Calabi–Yau algebras. When R is a preprojective algebra of type A and G is a finite subgroup of $D_n$ acting on R by automorphism, our main result shows that $\eta _{R,G}$ is an isomorphism if and only if G does not contain all of the reflections through a vertex.

Super orbifold theory

Let V be a simple vertex operator superalgebra and G a finite automorphism group of V containing the canonical automorphism σ such that VG is regular. It is proved that every irreducible VG-module occurs in an irreducible g-twisted V-module for some g∈G. Moreover, the quantum dimensions of irreducible VG-modules are determined, a global dimension formula for V in terms of twisted modules is obtained and a super quantum Galois theory is established. In addition the S-matrix of VG is computed.

G-twisted associative algebras of vertex operator superalgebras

Let V be a vertex operator superalgebra and G a finite automorphism group of V. Let σ be the order 2 automorphism of V associated with the superstructure of V. A sequence of associative algebras AG,n(V) are constructed to study the twisted representations of V for nonnegative n∈1T′Z where T′ is the order of group generated by G and σ. This result which generalizes many previous results on Zhu's algebras is then used to investigate the super orbifold theory. If V is a simple vertex operator superalgebra and S is a finite set of inequivalent irreducible twisted V-modules which is closed under the action of G, a duality theorem of Schur-Weyl type is obtained for the actions of certain finite dimensional semisimple associated algebra Aα(G,S) and VG on the direct sum of twisted V-modules in S. In particular, for any g∈G every irreducible g-twisted V-module is completely reducible VG-module.

On the geometry of K3 surfaces with finite automorphism group and no elliptic fibrations

Nikulin [On the quotient groups of the automorphism groups of hyperbolic forms by the subgroups generated by 2 reflections, J. Soviet. Math. 22 (1983) 1401–1476; Surfaces of type [Formula: see text] with a finite automorphism group and a Picard group of rank three, Proc. Steklov Institute of Math. (3) (1985) 131–155] and Vinberg [Classification of 2-reflective hyperbolic lattices of rank 4, Trans. Moscow Math. Soc. (2007) 39–66] proved that there are only a finite number of lattices of rank [Formula: see text] that are the Néron–Severi lattice of projective [Formula: see text] surfaces with a finite automorphism group. The aim of this paper is to provide a more geometric description of such [Formula: see text] surfaces [Formula: see text], when these surfaces have moreover no elliptic fibrations. In that case, we show that such [Formula: see text] surface is either a quartic with special hyperplane sections or a double cover of the plane branched over a smooth sextic curve which has special tangencies properties with some lines, conics or cuspidal cubic curves. We then study the converse, i.e. if the geometric description we obtained characterizes these surfaces. In four cases, the description is sufficient, in each of the four other cases, there is exactly another one possibility which we study. We obtain that at least five moduli spaces of [Formula: see text] surfaces (among the eight we study) are unirational.

Residues of connections and the Chevalley–Weil formula for curves

Given a finite group of automorphisms of a compact Riemann surface, the Chevalley-Weil formula computes the character valued Euler characteristic of an equivariant line bundle. The goal of this article is to give a proof by computing using residues of a Gauss-Manin connection.

Smooth quotients of abelian surfaces by finite groups that fix the origin

Let $A$ be an abelian surface and let $G$ be a finite group of automorphisms of $A$ fixing the origin. Assume that the analytic representation of $G$ is irreducible. We give a classification of the pairs $(A,G)$ such that the quotient $A/G$ is smooth. In particular, we prove that $A=E^2$ with $E$ an elliptic curve and that $A/G\simeq\mathbb{P}^2$ in all cases. Moreover, for fixed $E$, there are only finitely many pairs $(E^2,G)$ up to isomorphism. This fills a small gap in the literature and completes the classification of smooth quotients of abelian varieties by finite groups fixing the origin started by the first two authors.

Small Quotient Minimal Log Discrepancies

We prove that for each positive integer n, there exists a positive number ϵn such that n-dimensional toric quotient singularities satisfy the ACC for MLDs on the interval (0,ϵn). In the course of the proof, we show a geometric Jordan property for finite automorphism groups of affine toric varieties.

K3 surfaces with maximal finite automorphism groups containing M 20

It was shown by Mukai that the maximum order of a finite group acting faithfully and symplectically on a K3 surface is $960$ and that the group is isomorphic to the group $M_{20}$. Then Kondo showed that the maximum order of a finite group acting faithfully on a K3 surface is $3\,840$ and this group contains the Mathieu group $M_{20}$ with index four. Kondo also showed that there is a unique K3 surface on which this group acts faithfully, which is the Kummer surface $\Km(E_i\times E_i)$. In this paper we describe two more K3 surfaces admitting a big finite automorphism group of order $1\,920$, both groups contains $M_{20}$ as a subgroup of index 2. We show moreover that these two groups and the two K3 surfaces are unique. This result was shown independently by S. Brandhorst and K. Hashimoto in a forthcoming paper, with the aim of classifying all the finite groups acting faithfully on K3 surfaces with maximal symplectic part.

Fibrations associated to smooth quotients of abelian varieties

Let A be a complex abelian variety and G a finite group of automorphisms of A fixing the origin such that A/G is smooth. The quotient A/G can be seen as a fibration over an abelian variety whose fibers are isomorphic to a product of projective spaces. We classify how the fibers are glued in the case when the fibers are isomorphic to a projective space and we prove that, in general, the quotient A/G is a fibered product of such fibrations.

Coble surfaces with finite automorphism group

We classify Coble surfaces with finite automorphism group in characteristic \(p\ge 0, p\ne 2\). There are exactly 9 isomorphism classes of such surfaces.

On the solvability of graded Novikov algebras

We show that the right ideal of a Novikov algebra generated by the square of a right nilpotent subalgebra is nilpotent. We also prove that a [Formula: see text]-graded Novikov algebra [Formula: see text] over a field [Formula: see text] with solvable [Formula: see text]-component [Formula: see text] is solvable, where [Formula: see text] is a finite additive abelean group and the characteristic of [Formula: see text] does not divide the order of the group [Formula: see text]. We also show that any Novikov algebra [Formula: see text] with a finite solvable group of automorphisms [Formula: see text] is solvable if the algebra of invariants [Formula: see text] is solvable.