Translator Disclaimer
2022 Infinitesimal automorphisms of algebraic varieties and vector fields on elliptic surfaces
Gebhard Martin
Algebra Number Theory 16(7): 1655-1704 (2022). DOI: 10.2140/ant.2022.16.1655


We give several results concerning the connected component AutX0 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:XC 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 AutX0μpn 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 AutX0 explicitly in many concrete cases.


Download Citation

Gebhard Martin. "Infinitesimal automorphisms of algebraic varieties and vector fields on elliptic surfaces." Algebra Number Theory 16 (7) 1655 - 1704, 2022.


Received: 27 February 2021; Revised: 12 September 2021; Accepted: 2 November 2021; Published: 2022
First available in Project Euclid: 26 October 2022

Digital Object Identifier: 10.2140/ant.2022.16.1655

Primary: 14G17 , 14J27 , 14J50

Keywords: automorphisms , elliptic surfaces , group schemes , positive characteristic , vector fields

Rights: Copyright © 2022 Mathematical Sciences Publishers


This article is only available to subscribers.
It is not available for individual sale.

Vol.16 • No. 7 • 2022
Back to Top