Vector fields and moduli of canonically polarized surfaces in positive characteristic

TL;DR

This paper explores the geometry of canonically polarized surfaces in positive characteristic, revealing conditions under which such surfaces are unirational and their moduli stacks are Deligne-Mumford.

Contribution

It provides an explicit function relating the characteristic to surface properties, and identifies classes of surfaces with well-behaved moduli in positive characteristic.

Findings

Surfaces with $K_X^2 < f(p)$ are unirational.

The order of the algebraic fundamental group is at most two.

Certain classes of surfaces have moduli stacks that are Deligne-Mumford.

Abstract

This paper investigates the geometry of smooth canonically polarized surfaces defined over a field of positive characteristic which have a nontrivial global vector field, and the implications that the existence of such surfaces has in the moduli problem of canonically polarized surfaces. In particular, an explicit real valued function f(x) is obtained such that if $X$ is a smooth canonically polarized surface defined over an algebraically closed field of characteristic p>0 such that $K_X^2 <f(p)$, then $X$ is unirational and the order of its algebraic fundamental group is at most two. As a consequence of this result, large classes of canonically polarized surfaces are identified whose moduli stack is Deligne-Mumford, a property that does not hold in general in positive characteristic.