The KSBA compactification of the moduli space of $$D_{1,6}$$-polarized Enriques surfaces

We describe a compactification by stable pairs (also known as KSBA compactification) of the 4-dimensional family of Enriques surfaces which arise as the $${\mathbb {Z}}_2^2$$ Z 2 2 -covers of the blow up of $${\mathbb {P}}^2$$ P 2 at three general points branched along a configuration of three pairs of lines. Up to a finite group action, we show that this compactification is isomorphic to the toric variety associated to the secondary polytope of the unit cube. We relate the KSBA compactification considered to the Baily–Borel compactification of the same family of Enriques surfaces. Part of the KSBA boundary has a toroidal behavior, another part is isomorphic to the Baily–Borel compactification, and what remains is a mixture of these two. We relate the stable pair compactification studied here with Looijenga’s semitoric compactifications.

Automorphism Groups of Certain Enriques Surfaces

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.

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.