Infinite Automorphism Group Research Articles

The non-degeneracy invariant of Brandhorst and Shimada’s families of Enriques surfaces

Brandhorst and Shimada described a large class of Enriques surfaces, called (\tau,\bar{\tau}) -generic, for which they gave generators for the automorphism groups and calculated the elliptic fibrations and the smooth rational curves up to automorphisms. In the present paper, we give lower bounds for the non-degeneracy invariant of such Enriques surfaces, we show that in most cases the invariant has generic value 10 , and we present the first known example of complex Enriques surface with infinite automorphism group and non-degeneracy invariant not equal to 10 .

K3 surface entropy and automorphism groups

We derive a characterization of the complex projective K3 surfaces which have automorphisms of positive entropy in terms 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.

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

For an Enriques surface S, the non-degeneracy invariant nd ( S ) 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 nd ( S ) . 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 nd ( S ) = 10 which are not general and with infinite automorphism group. We obtain lower bounds on nd ( S ) 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.

On the geometry of singular K3 surfaces with discriminant 3, 4 and 7

We give a construction of singular K3 surfaces with discriminant 3 and 4 as double coverings over the projective plane. Focusing on the similarities in their branching loci, we can generalize this construction, and obtain a three dimensional moduli space of certain K3 surfaces which admit infinite automorphism groups. Moreover, we show that these K3 surfaces are characterized in terms of the configuration of the singular fibres of a jacobian elliptic fibration and also in terms of periods.

K3 Surfaces of zero entropy admitting an elliptic fibration with only irreducible fibers

We classify complex K3 surfaces of zero entropy admitting an elliptic fibration with only irreducible fibers. These surfaces are characterized by the fact that they admit a unique elliptic fibration with infinite automorphism group. We furnish an explicit list of 32 Néron-Severi lattices corresponding to such surfaces. Incidentally, we are able to decide which of these 32 classes of surfaces admit a unique elliptic pencil. Finally, we prove that all K3 surfaces with Picard rank ≥19 and infinite automorphism group have positive entropy.

Del Pezzo surfaces with infinite automorphism groups

Del Pezzo surfaces with infinite automorphism groups

Some Examples of K3 Surfaces with Infinite Automorphism Group which Preserves an Elliptic Pencil

We give more detail to our examples in [1] of K3 surfaces over $$\mathbb C$$ which have an infinite automorphism group that preserves some elliptic pencil of the K3 surface.

Central stability homology

We give a new categorical way to construct the central stability homology of Putman and Sam and explain how it can be used in the context of representation stability and homological stability. In contrast to them, we cover categories with infinite automorphism groups. We also connect central stability homology to Randal-Williams and Wahl's work on homological stability. We also develop a criterion that implies that functors that are polynomial in the sense of Randal-Williams and Wahl are centrally stable in the sense of Putman.

Fano threefolds with infinite automorphism groups

We classify smooth Fano threefolds with infinite automorphism groups.

Fano threefolds with infinite automorphism groups

Мы классифицируем гладкие трехмерные многообразия Фано с бесконечными группами автоморфизмов. Библиография: 55 наименований.

Elliptic fibrations on K3 surfaces and Salem numbers of maximal degree

We study the maximal Salem degree of automorphisms of K3 surfaces via elliptic fibrations. In particular, we establish a characterization of such maximum in terms of elliptic fibrations with infinite automorphism groups. As an application, we show that any supersingular K3 surface in odd characteristic has an automorphism the entropy of which is the natural logarithm of a Salem number of degree 22.

Hilbert schemes of lines and conics and automorphism groups of Fano threefolds

We discuss various results on Hilbert schemes of lines and conics and automorphism groups of smooth Fano threefolds with Picard rank 1. Besides a general review of facts well known to experts, the paper contains some new results, for instance, we give a description of the Hilbert scheme of conics on any smooth Fano threefold of index 1 and genus 10. We also show that the action of the automorphism group of a Fano threefold $X$ of index 2 (respectively, 1) on an irreducible component of its Hilbert scheme of lines (respectively, conics) is faithful if the anticanonical class of $X$ is very ample with a possible exception of several explicit cases. We use these faithfulness results to prove finiteness of the automorphism groups of most Fano threefolds and classify explicitly all Fano threefolds with infinite automorphism group. We also discuss a derived category point of view on the Hilbert schemes of lines and conics, and use this approach to identify some of them.

Elliptic Fibrations on K3 Surfaces

AbstractThis paper consists mainly of a review and applications of our old results relating to the title. We discuss how many elliptic fibrations and elliptic fibrations with infinite automorphism groups (or Mordell–Weil groups) an algebraic K3 surface over an algebraically closed field can have. As examples of applications of the same ideas, we also consider K3 surfaces with exotic structures: with a finite number of non-singular rational curves, with a finite number of Enriques involutions, and with naturally arithmetic automorphism groups.

Canonical vector heights on $K3$ surfaces – A nonexistence result

A. Baragar introduced a canonical vector height on a K3 surface X defined over a number field, and showed its existence if X has Picard rank two with infinite automorphism group. In another paper, A. Baragar and R. van Lujik performed numerical computation on certain K3 surfaces with Picard rank three, which strongly suggests that, in general, a canonical vector height does not exist. In this note, we prove this last assertion. We compare the set of periodic points of one automorphism with another on certain K3 surfaces.

Calabi-Yau threefolds with infinitely many divisorial contractions

We study Calabi-Yau 3-folds with infinitely many divisorial contractions. We also suggest a method to describe Calabi-Yau 3-folds with the infinite automorphism group.

ON THE CLASSIFICATION OF $4$ -DIMENSIONAL QUADRATIC DIVISION ALGEBRAS OVER SQUARE-ORDERED FIELDS

A square-ordered field, also called a Hilbert field of type (A), is understood to be an ordered field all of whose positive elements are squares. The problem of classifying, up to isomorphism, all 4-dimensional quadratic division algebras over a square-ordered field k is shown to be equivalent to the problem of finding normal forms for all pairs (X, Y) of 3 × 3 matrices over k, X being antisymmetric and Y being positive definite, under simultaneous conjugation by SO3(k). A solution is derived for the subproblem of this matrix pair problem defined by requiring Y+Yt to be orthogonally diagonalizable. The classifying list is given in terms of a 9-parameter family of configurations in k3,\sformed by a pair of points and an ellipsoid in normal position. Each 4-dimensional quadratic division algebra A over a square-ordered field k is shown to determine, uniquely up to sign, a self-adjoint linear endomorphism α of its purely imaginary hyperplane. Calling A diagonalizable in case α is orthogonally diagonalizable, the achieved solution of the matrix pair subproblem yields a full classification of all diagonalizable 4-dimensional quadratic division k-algebras. This generalizes earlier results of both Hefendehl-Hebeker who classified, over Hilbert fields, those 4-dimensional quadratic division algebras having infinite automorphism group, and Dieterich, who achieved a full classification of all real 4-dimensional quadratic division algebras. Finally, the paper describes explicitly how Hefendehl-Hebeker's classifying list, given in terms of a 4-parameter family of pairs of definite 3 × 3 matrices over k, embeds into the classifying list of configurations. The image of this embedding turns out to coincide with the sublist of the list formed by all non-generic configurations.

On the complete classification of extremal log Enriques surfaces

We show that there are exactly, up to isomorphisms, seven extremal log Enriques surfaces Z and construct all of them; among them types D_{19} and A_{19} have been shown of certain uniqueness by M. Reid. We also prove that the (degree 3 or 2) canonical covering of each of these seven Z has either X_3 or X_4 as its minimal resolution. Here X_3 (resp. X_4) is the unique K3 surface with Picard number 20 and discriminant 3 (resp. 4), which are called the most algebraic K3 surfaces by Vinberg and have infinite automorphism groups (by Shioda-Inose and Vinberg).

Rational points on K3 surfaces: A new canonical height

A fundamental tenet of Diophantine Geometry is that the geometric properties of an algebraic variety should determine its basic arithmetic properties. This is certainly true for curves, where the sign of the Euler characteristic of C determines whether the set of rational points on C is finite (X (C) 0). For higher dimensional varieties there are some precise conjectures due to Bombieri, Lang, and Vojta [15] which predict when the rational points on a variety should be finite or degenerate (i.e. not Zariski dense), and some conjectures of Manin et al. [-2, 5] on the distribution of rational points in those cases when they are Zariski dense. But except for abelian varieties, their subvarieties, and some Fano varieties (varieties for which the anticanonical bundle is ample), there are very few general theorems. In this paper we will study the rational points on a certain class of K 3 surfaces defined over a number field K. The moduli space of marked algebraic K3 surfaces is a countable union of 19 dimensional quasi-projective varieties. We are going to look at the 18 dimensional family studied by Wehler 1-17]. Wehler's family consists of K3 surfaces S whose automorphism group Aut(S) contains a subgroup d isomorphic to the free product 7/,2"~ 2 of two cyclic groups of order 2. We will use the geometric information provided by this infinite automorphism group to study the K-rat ional points on S. For any point PeS, we can look at the orbit of P under the action of d,

Rational Surfaces with Infinite Automorphism Group and No Antipluricanonical Curve

Rational Surfaces with Infinite Automorphism Group and No Antipluricanonical Curve

Rational surfaces with infinite automorphism group and no antipluricanonical curve

Counterexamples are given to a conjecture communicated to me by I. Dolgachev and E. Looijenga of M. Gizatullin that every rational surface with an infinite automorphism group should have an antipluricanonical curve.