On the projectivity of the moduli space of stable surfaces in characteristic p>5

TL;DR

This paper proves the projectivity of subspaces of the moduli space of stable surfaces in characteristic p>5, advancing understanding of their geometric properties and implications for algebraic geometry.

Contribution

It establishes the projectivity of the moduli space of stable surfaces over fields of characteristic p>5, under certain conjectural conditions.

Findings

Proper subspaces of the moduli space are projective.

The moduli space itself is projective over Z[1/30] under conjectural assumptions.

Provides new insights into the geometry of stable surfaces in positive characteristic.

Abstract

We prove that every proper subspace of the moduli space of stable surfaces with fixed volume over an algebraically closed field of characteristic p>5 is projective. As a consequence we also deduce that the same moduli space is projective over Z[1/30] modulo two conjectural local properties of the moduli functor.