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Infinitesimal automorphisms of algebraic varieties and vector fields on elliptic surfaces

Gebhard Martin

Vol. 16 (2022), No. 7, 1655–1704
Abstract

We give several results concerning the connected component Aut X0 of the automorphism scheme of a proper variety X over a field, such as its behavior with respect to birational modifications, normalization, restrictions to closed subschemes and deformations. Then, we apply our results to study the automorphism scheme of not necessarily Jacobian elliptic surfaces f : X C over algebraically closed fields, generalizing work of Rudakov and Shafarevich, while giving counterexamples to some of their statements. We bound the dimension h0(X,TX) of the space of global vector fields on an elliptic surface X if the generic fiber of f is ordinary or if f admits no multiple fibers, and show that, without these assumptions, the number h0(X,TX) can be arbitrarily large for any base curve C and any field of positive characteristic. If f is not isotrivial, we prove that Aut X0 μpn and give a bound on n in terms of the genus of C and the multiplicity of multiple fibers of f. As a corollary, we reprove the nonexistence of global vector fields on K3 surfaces and calculate the connected component of the automorphism scheme of a generic supersingular Enriques surface in characteristic 2. Finally, we present additional results on horizontal and vertical group scheme actions on elliptic surfaces which can be applied to determine Aut X0 explicitly in many concrete cases.

Keywords
elliptic surfaces, automorphisms, group schemes, vector fields, positive characteristic
Mathematical Subject Classification
Primary: 14G17, 14J27, 14J50
Milestones
Received: 27 February 2021
Revised: 12 September 2021
Accepted: 2 November 2021
Published: 16 October 2022
Authors
Gebhard Martin
Mathematisches Institut der Universität Bonn
Universität Bonn
Bonn
Germany