Virus infections, Corona surfaces, and extra components in the moduli space of stable surfaces

TL;DR

This paper introduces Corona surfaces, a class of non-smoothable stable surfaces with diverse invariants, revealing that the moduli space of stable surfaces has multiple connected components distinguished by the second plurigenus, exceeding the complexity of the Gieseker moduli space.

Contribution

It constructs examples of non-smoothable stable surfaces called Corona surfaces, expanding understanding of the structure of the moduli space of stable surfaces.

Findings

Corona surfaces have invariants covering all smooth minimal surfaces of general type.

The moduli space of stable surfaces has at least k-l+2 connected components.

There are more irreducible components in the stable surfaces moduli space than in the Gieseker moduli space.

Abstract

We construct examples of non-smoothable stable surfaces, that we call Corona surfaces, with invariants in a wide range comprising all possible invariants of smooth minimal surfaces of general type but non-standard second plurigenus. We deduce that the moduli space of stable surfaces $\overline{\mathfrak M}_{k,l}$ has least $k-l+2$ connected components distinguished by the second plurigenus and, in particular, it always has more irreducible components than the Gieseker moduli space with the same invariants.