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.
We study automorphism groups and birational automorphism groups of compact complex surfaces. We show that the automorphism group of such surface $X$ is always Jordan, and the birational automorphism group is Jordan unless $X$ is birational to a product of an elliptic and a rational curve.
We classify del Pezzo surfaces with Du Val singularities that have infinite automorphism groups, and describe the connected components of their automorphisms groups.
We show that the automorphism group of a certain subclass of smooth Gizatullin surfaces with a distinguished and rigid extended divisor is generated by automorphisms of A1-fibrations. Moreover, such surfaces provide examples of smooth Gizatullin surfaces with a non-transitive action of the automorphism group. Thus, they represent counterexamples to Gizatullin's conjecture. For such surfaces we give explicit orbits of the natural action of the automorphism group in some special cases. Further, we present their automorphism groups as amalgamated products of two subgroups.
We calculate the automorphism group of certain Enriques surfaces. The Enriques surfaces that we investigate include very general $n$-nodal Enriques surfaces and very general cuspidal Enriques surfaces. We also describe the action of the automorphism group on the set of smooth rational curves and on the set of elliptic fibrations.
We will show that there is a smooth complex projective surface, birational to some Enriques surface, such that the automorphism group is discrete but not finitely generated.
We classify Coble surfaces with finite automorphism group in arbitrary characteristic not equal to 2. There are exactly 9 isomorphism classes of such surfaces.
We develop technics of birational geometry to study automorphisms of affine surfaces admitting many distinct rational fibrations, with a particular focus on the interactions between automorphisms and these fibrations. In particular, we associate to each surface $S$ of this type a graph encoding equivalence classes of rational fibrations from which it is possible to decide for instance if the automorphism group of $S$ is generated by automorphisms preserving these fibrations.
In this note we show that if the automorphism group of a normal affine surface $S$ is isomorphic to the automorphism group of a Danielewski surface, then $S$ is isomorphic to a Danielewski surface.
We present finite sets of generators of the full automorphism groups of three singular K3 surfaces, on which the alternating group of degree 6 acts symplectically. We also present a finite set of generators of the full automorphism group of an associated Enriques surface, on which the Mathieu group M_{10} acts.
In this paper we classify all Riemann surfaces having a large abelian group of automorphisms, that is having an abelian group of automorphism of order strictly bigger then $4(g-1)$, where $g$ denotes as usual the genus of the Riemann surface.
We derive a characterization of the complex projective K3 surfaces which have automorphisms of positive entropy in term of their Néron-Severi lattices. Along the way, we classify the projective K3 surfaces of zero entropy with infinite automorphism groups and we determine the projective K3 surfaces of Picard number at least five with almost abelian automorphism groups, which gives an answer to a long standing question of Nikulin.
This paper concerns K3 surfaces with automorphisms of order 11 in arbitrary characteristic. Specifically we study the wild case and prove that a general such surface in characteristic 11 has Picard number 2. We also construct K3 surfaces with an automorphism of order 11 in every characteristic, and supersingular K3 surfaces whenever possible.
We prove some general results concerning the size of the group of automorphisms of the Douady space of points on a surface. We then study some properties of the automorphisms coming from an automorphism of the surface, in particular their action on the cohomology and the classification of their fixed points.
We determine the minimum positive entropy of complex Enriques surface automorphisms. This together with McMullen's work completes the determination of the minimum positive entropy of complex surface automorphisms in each class of Enriques-Kodaira classification of complex surfaces.
We classify finite groups acting by birational transformations of a non-trivial Severi--Brauer surface over a field of characteristc zero that are not conjugate to subgroups of the automorphism group. Also, we show that the automorphism group of a smooth cubic surface over a field $K$ of characteristic zero that has no $K$-points is abelian, and find a sharp bound for the Jordan constants of birational automorphism groups of such cubic surfaces.
We determine all possible orders of automorphisms of complex K3 surfaces. A positive integer N is the order of an automorphism of a complex K3 surface if and only if $ϕ(N) \leq 20$ where $ϕ$ is the Euler function.
We treat non-symplectic automorphisms on $K3$ surfaces which act trivially on the Néron-Severi lattice. In this paper, we classify non-symplectic automorphisms of prime-power order, especially 2-power order on $K3$ surfaces, i.e., we describe their fixed locus.
A few facts concerning the phrase "the automorphism groups become larger at special points of the moduli of K3 surfaces" are presented. It is also shown that the automorphism groups are of infinite order over a dense subset in any one-dimensional non-trivial family of projective K3 surfaces.
Complex Enriques surfaces with a finite group of automorphisms are classified into seven types. In this paper, we determine which types of such Enriques surfaces exist in characteristic 2. In particular we give a one dimensional family of classical and supersingular Enriques surfaces with the automorphism group isomorphic to the symmetric group of degree five.
We prove that, for an Enriques surface in odd characteristic, the automorphism group is finitely generated and it acts on the effective nef cone with a rational polyhedral fundamental domain. We also construct a smooth projective surface in odd characteristic which is birational to an Enriques surface and whose automorphism group is discrete but not finitely generated.
We study the groups of automorphisms of rational algebraic surfaces that admit a relatively minimal pencil of curves of arithmetic genus one over an algebraically closed field of arbitrary characteristic. In particular, we classify such surfaces that admit non-trivial automorphisms that act trivially on the Picard group. As an application, we classify classical Enriques surfaces in characteristic $2$ that admit non-trivial numerically trivial automorphisms.
In this paper we study the automorphisms group of some K3 surfaces which are double covers of the projective plane ramified over a smooth sextic plane curve. More precisely, we study some particlar case of a K3 surface of Picard rank two.
We classify supersingular and classical Enriques surfaces with finite automorphism group in characteristic 2 into 8 types according to their dual graphs of all $(-2)$-curves (nonsigular rational curves). We give examples of these Enriques surfaces together with their canonical coverings. It follows that the classification of all Enriques surfaces with finite automorphism group in any characteristics has been finished.
We give more details to our examples in [9] of K3 surfaces over C such that they have infinite automorphism group but it preserves some elliptic pencil of the K3
We extend to arbitrary characteristic some known results about automorphisms of complex Enriques surfaces that act trivially on the cohomology or the cohomology modulo torsion.
For a relatively minimal surface fibration $f: X\to C$, the equivariant automorphism group of $f$ is, roughly speaking, the group of automorphisms of $X$ preserving the fibration structure. We present a classification of such fibrations of fibre genus $g\ge 1$ with smooth generic fibre over an arbitrary algebraically closed field $\mathbf{k}$ whose equivariant automorphism group is infinite.
In this paper we investigate when the generic member of a family of K3 surfaces admitting a non--symplectic automorphism of finite order admits also a symplectic automorphism of the same order. We give a complete answer to this question if the order of the automorphism is a prime number and we provide several examples and partial results otherwise. Moreover we prove that, under certain conditions, a K3 surface admitting a non--symplectic automorphism of prime odd order, $p$, also admits a non--symplectic automorphism of order $2p$. This generalizes a previous result by J. Dillies for $p=3$.
In this note we exhibit explicit automorphisms of maximal Salem degree 22 on the supersingular K3 surface of Artin invariant one for all primes p congruent 3 mod 4 in a systematic way. Automorphisms of Salem degree 22 do not lift to any characteristic zero model.
We determine the automorphism group of the Hilbert scheme of two points on a generic projective K3 surface of any polarization. We obtain in particular new examples of Hilbert schemes of points having non-natural non-symplectic automorphisms. The existence of these automorphisms depends on solutions of Pell's equation.
This paper is a survey about $K3$ surfaces with an automorphism and log rational surfaces, in particular, log del Pezzo surfaces and log Enriques surfaces. It is also a reproduction on my talk at "Mathematical structures of integrable systems and their applications" held at Research Institute for Mathematical Sciences in September 2018.
We classify all possible automorphism groups of smooth cubic surfaces over an algebraically closed field of arbitrary characteristic. As an intermediate step we also classify automorphism groups of quartic del Pezzo surfaces. We show that the moduli space of smooth cubic surfaces is rational in every characteristic, determine the dimensions of the strata admitting each possible isomorphism class of automorphism group, and find explicit normal forms in each case. Finally, we completely characterize when a smooth cubic surface in positive characteristic, together with a group action, can be lift…
For every p >= 5, we determine all Z_p-invariant nonsingular quartic surfaces in the three dimensional projective space over an algebraically closed field of characteristic zero. In some cases, we also determine their full projective automorphism groups.
We classify all the K3 surfaces which are minimal models of the quotient of the product of two curves $C_1\times C_2$ by the diagonal action of either the group $\Z/p\Z$ or the group $\Z/2p\Z$. These K3 surfaces admit a non-symplectic automorphism of order $p$ induced by an automorphism of one of the curves $C_1$ or $C_2$. We prove that most of the K3 surfaces admitting a non-symplectic automorphism of order $p$ (and in fact a maximal irreducible component of the moduli space of K3 surfaces with a non-symplectic automorphism of order $p$) are obtained in this way.\\ In addition, we show that o…
We characterize simple complex abelian varieties and simple abelian surfaces in terms of primitivity of translation automorphisms. Applying this together with a result due to Diller and Favre, we then classify all primitive birational automorphisms with trivial first dynamical degree of smooth projective surfaces over an algebraically closed field of any characteristic.
We classify the automorphism group of minimal surfaces of general type with $K_S^2 = 1$ and $ρ_g = 2$. Furthermore, we show that the order of the automorphism group is bounded above by 200 and can only have prime factors $p \leq 31$ with $p \neq 29$.
In characteristic $p=0$ or $p>5$, we show that a K3 surface with an order 60 automorphism is unique up to isomorphism. As a consequence, we characterize the supersingular K3 surface with Artin invariant 1 in characteristic $p=11$ (mod 12) by a cyclic symmetry of order 60.
Recently Oguiso showed the existence of K3 surfaces that admit a fixed point free automorphism of positive entropy. The K3 surfaces used by Oguiso have a particular rank two Picard lattice. We show, using results of Beauville, that these surfaces are therefore determinantal quartic surfaces. Long ago, Cayley constructed an automorphism of such determinantal surfaces. We show that Cayley's automorphism coincides with Oguiso's free automorphism. We also exhibit an explicit example of a determinantal quartic whose Picard lattice has exactly rank two and for which we thus have an explicit descript…
We give a short proof that every supersingular K3 surface (except possibly in characteristic $2$ with Artin invariant $σ=10$) has an automorphism of Salem degree 22. In particular an infinite subgroup of the automorphism group does not lift to characteristic zero. The proof relies on the case $σ=1$ and the cone conjecture for K3 surfaces.
In this article we study compact Riemann surfaces with a non-large group of automorphisms of maximal order; namely, compact Riemann surfaces of genus $g$ with a group of automorphisms of order $4g-4.$ Under the assumption that $g-1$ is prime, we provide a complete classification of them and determine isogeny decompositions of the corresponding Jacobian varieties.
It follows from an observation of A. Coble in 1919 that the automorphism group of an unnodal Enriques surface contains the $2$-congruence subgroup of the Weyl group of the $E_{10}$-lattice. In this article, we determine how much bigger the automorphism group of an unnodal Enriques surface can be. Furthermore, we show that the automorphism group is in fact equal to the $2$-congruence subgroup for generic Enriques surfaces in arbitrary characteristic (under the additional assumption that the Enriques surface is ordinary if the characteristic is $2$), improving the corresponding result of W. Bart…
In this article, we focus on a new perspective of automorphisms of complex 2-tori, reviewing previous works from a lattice-theoretic point of view. In particular, we give a classification of families of symplectic and non-symplectic automorphisms of prime order.
We classify the bi-canonical representations of finite automorphisms on Enriques surfaces. There are three types of non-trivial cases and examples are given explicitly by Horikawa models. In particular, finite non-semi-symplectic automorphisms exist only in orders 4 and 8. One corollary is that any finite cyclic subgroup in the automorphism group of an Enriques surface has order 1, 2, 3, 4, 5, 6, 8. Moreover, for two of the three types, a uniqueness theorem for maximal-dimensional families is given.
We classify primitive non-symplectic automorphisms of order 6 on K3 surfaces. We show how their study can be reduced to the study of non-symplectic automorphisms of order 3 and to a local analysis of the fixed loci. In particular, we determine the possible fixed loci and show that when the Picard lattice is fixed, K3 surfaces come in mirror pairs.
In this paper we study K3 surfaces with a non-symplectic automorphism of order 3. In particular, we classify the topological structure of the fixed locus of such automorphisms and we show that it determines the action on cohomology. This allows us to describe the structure of the moduli space and to show that it has three irreducible components.
