The structure of Frobenius kernels for automorphism group schemes

TL;DR

This paper investigates the structure of Frobenius kernels in automorphism group schemes of surfaces of general type over fields of positive characteristic, revealing limited possibilities and introducing a new invariant called foliation rank.

Contribution

It provides new structural results for Frobenius kernels, utilizing properties of the Witt algebra and twisted forms, applicable to a broad class of schemes.

Findings

Frobenius kernels have surprisingly few structural possibilities.

The results depend on the foliation rank, a new numerical invariant.

The findings extend to arbitrary proper integral schemes under certain conditions.

Abstract

We establish structure results for Frobenius kernels of automorphism group schemes for surfaces of general type in positive characteristics. It turns out that there are surprisingly few possibilities. This relies on properties of the famous Witt algebra, which is a simple Lie algebra without finite-dimensional counterpart over the complex numbers, together with is twisted forms. The result actually holds true for arbitrary proper integral schemes under the assumption that the Frobenius kernel has large isotropy group at the generic point. This property is measured by a new numerical invariant called the foliation rank.