Infinitesimal Automorphisms of Algebraic Varieties
Final Report Abstract
The purpose of this project was to achieve a better understanding of automorphism schemes of smooth projective varieties in positive characteristic p > 0, starting with the case of surfaces. More specifically, I wanted to study the cases where this automorphism scheme is non-reduced. Due to Cartier’s theorem, this is a phenomenon that can only occur in positive characteristic. In the case of surfaces of negative Kodaira dimension, Claudia Stadlmayr and I achieved a complete classification of weak del Pezzo surfaces with nontrivial global vector fields in arbitrary characteristic in the article “Weak del Pezzo surfaces with global vector fields”. In particular, we showed that there exist weak del Pezzo surfaces with non-reduced automorphism schemes if and only if p ∈ {2, 3}. For surfaces of Kodaira dimension 0, I determined the automorphism schemes of (quasi-)bielliptic surfaces in arbitrary characteristic in the article “Automorphism schemes of (quasi-)bielliptic surfaces”. This generalizes work of Bennett and Miranda and corrects a small gap in their classification. Again, non-reduced automorphism schemes occur if and only if p ∈ {2, 3}. Since automorphism schemes of blow-ups of K3 surfaces and Abelian surfaces are reduced in all characteristics and the same holds for blow-ups of Enriques surfaces if p ≠ 2, this article completes our picture of the schemestructure of the automorphism schemes of surfaces of Kodaira dimension 0 if p ≠ 2. Covering most surfaces of Kodaira dimension 1 (except for the quasielliptic surfaces in characteristic p ∈ {2, 3}), I studied automorphism schemes of elliptic surfaces in the article “Infinitesimal automorphisms of algebraic varieties and vector fields on elliptic surfaces”: In this article, I give bounds on the length of the automorphism scheme depending on cohomological invariants of the elliptic surface and construct examples showing that the automorphism scheme of an elliptic surface can be arbitrarily non-reduced (in the sense that the tangent dimension is unbounded while the dimension is bounded) in every positive characteristic. Applications include a new proof — following ideas of Rudakov and Shafarevich, and filling in some gaps in their proof in small characteristics — of the non-existence of non-trivial global vector fields on K3 surfaces as well as a proof of the fact that the automorphism scheme of a generic supersingular Enriques surface in characteristic 2 is µ2 . Furthermore, I hope that the general results on automorphism schemes of arbitrary proper varieties obtained in this article, such as a fixed point formula for linearly reductive group scheme actions and a study of the behaviour of automorphism schemes under deformations and birational modifications, will find further applications in the study of infinitesimal automorphisms of algebraic varieties.
Publications
- Infinitesimal automorphisms of algebraic varieties and vector fields on elliptic surfaces
Gebhard Martin
- “Automorphism schemes of (quasi-)bielliptic surfaces”, 15 pages, August 2020
Gebhard Martin
