Enriques surfaces with non-generic non-degeneracy

TL;DR

This paper investigates the non-degeneracy invariant of complex Enriques surfaces, proving it cannot increase under specialization and computing it for numerous families, revealing new examples with non-generic invariants.

Contribution

It establishes the non-increase of the non-degeneracy invariant under specialization and computes it for 155 families, identifying the first examples with invariant 9.

Findings

Non-degeneracy invariant cannot increase under specialization.

Computed invariant for 155 families of Enriques surfaces.

Discovered the first examples with invariant 9.

Abstract

We study the non-degeneracy invariant $\mathrm{nd}(Y)$ of complex Enriques surfaces in families. Our first main result shows that $\mathrm{nd}(Y)$ cannot increase under specialization. The second main result is the conclusion of the computation of the non-degeneracy invariant for the $155$ families of $(\tau,\overline{\tau})$-generic surfaces introduced by Brandhorst and Shimada. Of the previously known $144$ cases, only $3$ satisfy $\mathrm{nd}(Y)\neq10$, which is the non-degeneracy invariant of a general Enriques surface. The remaining $11$ families studied in this article also have non-generic non-degeneracy. To compute this, we produce upper bounds on $\mathrm{nd}(Y)$ by refining this invariant into two others: the Fano and Mukai non-degeneracy invariants, which are related to two different classes of projective realizations of Enriques surfaces. As a result, we find the first known examples of Enriques surfaces with $\mathrm{nd}(Y)=9$.