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Arithmetic and Geometry around hypergeometric functions, Progress in Mathematics, Birkhauser, , Maximal subgroups of the Mathieu group M23 and symmetric automorphisms of supersingular K3 surfaces,. A complex ball uniformization of the moduli space of cubic surfaces via periods of K3 surfaces,.

The moduli space of curves of genus 4 and Deligne-Mostow's complex reflection groups,. In characteristic 2 the definition of Enriques surfaces is modified: In characteristics other than 2 this is equivalent to the usual definition. There are now 3 families of Enriques surfaces:.

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## Failure of the Hasse principle for Enriques surfaces - ScienceDirect

In particular, we prove that unnodal Enriques surfaces are of Ulrich wild representation type with respect to an arbitrary multiple of a Fano polarization by constructing moduli spaces of stable Ulrich bundles of arbitrary rank and arbitrarily large dimension. In addition to having important connections to the study of deformation theory e. This complexity is called the representation type of the variety [ 25 ].

Varieties admitting only a finite number of indecomposable ACM sheaves up to twist and isomorphism are called of finite representation type and have been completely classified [ 29 ]: At the other extreme are varieties of wild representation type for which there exist families of non-isomorphic indecomposable ACM sheaves of arbitrarily large dimension.

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In higher dimension, we cannot expect such a simple trichotomy to hold. It is worth noting that the definition of the representation type of a variety is dependent on and sensitive to the specific projective embedding chosen, and indeed not all non-degenerate linearly normal subvarieties are ACM. For Enriques surfaces, which we discuss in this article, examples of such embeddings are given in [ 19 ].

In this article, we study a particular kind of ACM bundle whose associated graded module has the maximal number of generators.

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Although the algebraic counterpart of this phenomenon was first observed by Ulrich in [ 27 ], these so-called Ulrich sheaves on projective varieties were originally introduced in [ 15 ]. These equivalences are the content of [ 15 , Proposition 2. Although Ulrich sheaves are conjectured to exist on every projective variety [ 15 ], Ulrich bundles have only been constructed in a few cases: In this short note, we will study Ulrich bundles on a generic more specifically: We also formulate an attractive lattice theoretic conjecture that would ensure that Ulrich line bundles exist for all polarizations.

In Section 3 , we construct stable Ulrich bundles of any rank on an unnodal Enriques surface with moduli spaces of increasingly large dimension.

## Enriques surface

As a corollary, we show that with respect to multiples of a Fano polarization unnodal Enriques surfaces are of Ulrich wild representation type. That is, they are of wild representation type such that the indecomposable ACM bundles may be taken to be Ulrich, as in [ 17 , Definition 2]. Enriques surface to be acyclic, in the sense that all of its cohomology groups vanish. The following purely lattice theoretic conjecture is appealing but we cannot presently prove it. The particular case of Conjecture 2. We believe the conjecture is true in general, having checked it on a number of small examples.