Automorphisms of smooth canonically polarized surfaces in characteristic 2

TL;DR

This paper studies the automorphism schemes of smooth canonically polarized surfaces in characteristic 2, revealing conditions under which these schemes are non-smooth and describing the geometric properties of such surfaces.

Contribution

It provides new criteria for the smoothness of automorphism schemes and characterizes surfaces with non-smooth automorphism schemes in characteristic 2.

Findings

Surfaces with 1 ≤ K_X^2 ≤ 2 and non-smooth automorphism schemes are uniruled.

If K_X^2=1, the surface is simply connected, unirational, and has p_g ≤ 1.

Such surfaces are purely inseparable quotients of rational surfaces by rational vector fields.

Abstract

This paper investigates the structure of the automorphism scheme of a smooth canonically polarized surface $X$ defined over an algebraically closed field of characteristic 2. In particular it is investigated when Aut(X) is not smooth. This is a situation that appears only in positive characteristic and it is closely related to the structure of the moduli stack of canonically polarized surfaces. Restrictions on certain numerical invariants of X are obtained in order for Aut(X) to be smooth or not and information is provided about the structure of the component of Aut(X) containing the identity. In particular, it is shown that if X is a smooth canonically polarized surface with 1\leq K_X^2 \leq 2 with non smooth automorphism scheme, then X is uniruled. Moreover, if K_X^2=1, then X is simply connected, unirational and p_g(X) \leq 1. Moreover, X is the purely inseparable quotient of a rational surface by a rational vector field.