Enriques surfaces and Jacobian elliptic K3 surfaces

This paper proposes a new geometric construction of Enriques surfaces. Its starting point are K3 surfaces with Jacobian elliptic fibration which arise from rational elliptic surfaces by a quadratic base change. The Enriques surfaces obtained in this way are characterised by elliptic fibrations with a rational curve as bisection which splits into two sections on the covering K3 surface. The construction has applications to the study of Enriques surfaces with specific automorphisms. It also allows us to answer a question of Beauville about Enriques surfaces whose Brauer groups show an exceptional behaviour. In a forthcoming paper, we will study arithmetic consequences of our construction.     

On Lifting Fibrations of Genus One

We investigate the problem of lifting fibrations of genus one on algebraic surfaces of Kodaira dimension zero. We prove that fibrations on the following surfaces lift: Enriques surfaces, K3 surfaces covering Enriques surfaces, certain hyperelliptic, and quasi-hyperelliptic surfaces.     

Monodromy groups of real Enriques surfaces

We compute the monodromy groups of real Enriques surfaces of hyperbolic type. The principal tools are the deformation classification of such surfaces and a modified version of Donaldson?s trick, relating real Enriques surfaces and real rational surfaces.     

Finite Abelian Groups of K3 Surfaces with Smooth Quotient

The quotient space of a K3 surface by a finite group is an Enriques surface or a rational surface if it is smooth. Finite groups where the quotient space are Enriques surfaces are known. In this paper, by analyzing effective divisors on smooth rational surfaces, the author will study finite groups which act faithfully on K3 surfaces such that the quotient space are smooth. In particular, he will completely determine effective divisors on Hirzebruch surfaces such that there is a finite Abelian co...     

On automorphisms of Enriques surfaces and their entropy

Consider an arbitrary automorphism of an Enriques surface with its lift to the covering K3 surface. We prove a bound of the order of the lift acting on the anti‐invariant cohomology sublattice of the Enriques involution. We use it to obtain some mod 2 constraint on the original automorphism. As an application, we give a necessary condition for Salem numbers to be dynamical degrees on Enriques surfaces and obtain a new lower bound on the minimal value. In the Appendix, we give a complete list of Salem numbers that potentially could be the minimal dynamical degree on Enriques surfaces and for which the existence of geometric automorphisms is unknown.     

Ulrich line bundles on Enriques surfaces with a polarization of degree four

In this paper, we prove the existence of an Enriques surface with a polarization of degree four with an Ulrich bundle of rank one. As a consequence, we prove that general polarized Enriques surfaces of degree four, with the same numerical polarization class, carry Ulrich line bundles.     

Higher dimensional Enriques varieties and automorphisms of generalized Kummer varieties

We define Enriques varieties as a higher dimensional generalization of Enriques surfaces and construct examples by using fixed point free automorphisms on generalized Kummer varieties. We also classify all automorphisms of generalized Kummer varieties that come from an automorphism of the underlying abelian surface.     

Algebraic cycles and topology of real Enriques surfaces

For a real Enriques surface Y we prove that every homology class in H₁(Y (R), Z/2) can be represented by a real algebraic curve if and only if all connected components of Y(R) are orientable. Furthermore, we give a characterization of real Enriques surfaces which are Galois-Maximal and/or Z-Galois-Maximal and we determine the Brauer group of any real Enriques surface Y.     

Classification of Enriques Log Surfaces with δ=1

The Enriques log surfaces with δ= 1 are classified.     

On nodal Enriques surfaces and quartic double solids

We consider the class of singular double coverings \(X \rightarrow {\mathbb {P}}^3\) ramified in the degeneration locus \(D\) of a family of 2-dimensional quadrics. These are precisely the quartic double solids constructed by Artin and Mumford as examples of unirational but nonrational conic bundles. With such a quartic surface \(D,\) one can associate an Enriques surface \(S\) which is the factor of the blowup of \(D\) by a natural involution acting without fixed points (such Enriques surfaces are known as nodal Enriques surfaces or Reye congruences). We show that the nontrivial part of the derived category of coherent sheaves on this Enriques surface \(S\) is equivalent to the nontrivial part of the derived category of a minimal resolution of singularities of \(X\).     

On the moduli space of real Enriques surfaces

We introduce a new invariant, Pontryagin-Viro form, of real algebraic surfaces. We evaluate it for real Enriques surfaces with non-negative minimal Euler characteristic of the components of the real part and prove that, when combined with the known topological invariants, it distinguishes the deformation types of such surfaces.     

Twisted Stability and Fourier–Mukai Transform I

In this paper, we consider the preservation of stability by using the notion of twisted stability. As applications, (1) we show that moduli spaces of stable sheaves on K3 and abelian surfaces are irreducible and (2) we compute Hodge polynomials of some moduli spaces of stable sheaves on Enriques surfaces.     

On Vorontsov's Theorem on K3 surfaces with non-symplectic group actions

We shall give a proof for Vorontsov's Theorem and apply this to classify log Enriques surfaces with large prime canonical index.

     

On positivity of line bundles on Enriques surfaces

We study linear systems on Enriques surfaces. We prove rationality of Seshadri constants of ample line bundles on Enriques surfaces and provide lower bounds on these numbers. We show the nonexistence of -very ample line bundles on Enriques surfaces of degree for , thus answering an old question of Ballico and Sommese.

     

Projective models of Enriques surfaces in scrolls

We study Cossec's ? ‐function, which is defined for divisors with positive self‐intersection on an Enriques surface. In this paper we study the existence of pairs (C ², ? (C )) with C an irreducible curve. The ? ‐function gives in a natural way scrolls containing Enriques surfaces. We compute scroll types to some of these scrolls. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Classification of Logarithmic Enriques Surfaces with δ=2

We classify logarithmic Enriques surfaces with = 2     

Comment on: On the irreducibility of the Severi variety of nodal curves in a smooth surface,by E. Ballico

Archiv der Mathematik - In this short note, I point out that results of Ballico and Kool–Shende–Thomas together imply that on K3, Enriques, and Abelian surfaces, if L is a very ample...     

Exceptional vector bundles on Enriques surfaces

The main purpose of this paper is to study exceptional vector bundles on Enriques surfaces. Translated fromMatematicheskie Zametki, Vol. 61, No. 6, pp. 825–834, June, 1997.     

Severi varieties on blow-ups of the symmetric square of an elliptic curve

We prove that certain Severi varieties of nodal curves of positive genus on general blow-ups of the twofold symmetric product of a general elliptic curve are nonempty and smooth of the expected dimension. This result, besides its intrinsic value, is an important preliminary step for the proof of nonemptiness of Severi varieties on general Enriques surfaces.