Higher-dimensional Calabi–Yau varieties with dense sets of rational points

We construct higher-dimensional Calabi–Yau varieties defined over a given number field with Zariski dense sets of rational points. We give two elementary constructions in arbitrary dimensions as well as another construction in dimension three which involves certain Calabi–Yau threefolds containing an Enriques surface. The constructions also show that potential density holds for (sufficiently) general members of the families.

Rank one sheaves over quaternion algebras on Enriques surfaces

Let X be an Enriques surface over the field of complex numbers. We prove that there exists a nontrivial quaternion algebra 𝓐 on X. Then we study the moduli scheme of torsion free 𝓐-modules of rank one. Finally we prove that this moduli scheme is an étale double cover of a Lagrangian subscheme in the corresponding moduli scheme on the associated covering K3 surface.

Twelve Rational curves on Enriques surfaces

Given $$d\in {\mathbb {N}}$$ d ∈ N , we prove that any polarized Enriques surface (over any field k of characteristic $$p \ne 2$$ p ≠ 2 or with a smooth K3 cover) of degree greater than $$12d^2$$ 12 d 2 contains at most 12 rational curves of degree at most d. For $$d>2$$ d > 2 , we construct examples of Enriques surfaces of high degree that contain exactly 12 rational degree-d curves.

On an Enriques Surface Associated With a Quartic Hessian Surface

Let $Y$ be a complex Enriques surface whose universal cover $X$ is birational to a general quartic Hessian surface. Using the result on the automorphism group of $X$ due to Dolgachev and Keum, we obtain a finite presentation of the automorphism group of $Y$. The list of elliptic fibrations on $Y$ and the list of combinations of rational double points that can appear on a surface birational to $Y$ are presented. As an application, a set of generators of the automorphism group of the generic Enriques surface is calculated explicitly.

Special Ulrich bundles on non-special surfaces with pg = q = 0

Let [Formula: see text] be a surface with [Formula: see text] and endowed with a very ample line bundle [Formula: see text] such that [Formula: see text]. We show that [Formula: see text] supports special (often stable) Ulrich bundles of rank [Formula: see text], extending a recent result by A. Beauville. Moreover, we show that such an [Formula: see text] supports families of dimension [Formula: see text] of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrary large [Formula: see text] except for very few cases. We also show that the same is true for each linearly normal non-special surface with [Formula: see text] in [Formula: see text] of degree at least [Formula: see text], Enriques surface and anticanonical rational surface.