Narrow Results

Reflection walls of certain primitive vectors in the anti-invariant sublattice of the K3 lattice define Heegner divisors in the period space of Enriques surfaces. We show that depending on the norm of these primitive vectors, these Heegner divisors are either irreducible or have two irreducible components. The two components are obtained as the walls orthogonal to primitive vectors of the same norm but of different type as ordinary or characteristic.

Let $S$ be a surface with $p_g(S)=q(S)=0$ and endowed with a very ample line bundle ${{𝒪}}_S(h)$ such that $h^1(S,{{𝒪}}_S(h))=0$. We show that $S$ supports special (often stable) Ulrich bundles of rank $2$, extending a recent result by A. Beauville. Moreover, we show that such an $S$ supports families of dimension $p$ of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrary large $p$ except for very few cases. We also show that the same is true for each linearly normal non-special surface with $p_g(S)=q(S)=0$ in ${\mathbb{P}}^4$ of degree at least $4$, Enriques surface and anticanonical rational surface.