Enriques surfaces with non-generic non-degeneracy

2025/12/21 by Riccardo Moschetti, Moschetti, Riccardo, Franco Rota +3
Mathematics · #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Homotopy and Cohomology in Algebraic Topology

paper · doi:10.48550/arxiv.2512.18812

Abstract

We study the non-degeneracy invariant nd(Y) of complex Enriques surfaces in families. Our first main result shows that 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 (τ,τ)-generic surfaces introduced by Brandhorst and Shimada. Of the previously known 144 cases, only 3 satisfy nd(Y)≠10, 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 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 nd(Y)=9.

Citations

• Enriques surfaces of zero entropy
• The non-degeneracy invariant of Brandhorst and Shimada's families of Enriques surfaces
• Mordell-Weil groups and automorphism groups of elliptic K3 surfaces
• Automorphism groups of certain Enriques surfaces
• Borcherds' method for Enriques surfaces
• Enriques surfaces with eight nodes

Related

• The non-degeneracy invariant of Brandhorst and Shimada's families of Enriques surfaces
• A Refined Derived Torelli Theorem for Enriques surfaces, II: the non-generic case
• A brief introduction to Enriques surfaces
• A note on the derived category of Enriques surfaces in characteristic 2
• A 1-dimensional family of Enriques surfaces in characteristic 2 covered by the supersingular K3 surface with Artin invariant 1