We give several results concerning the connected component of the automorphism scheme of a proper variety 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 over algebraically closed fields, generalizing work of Rudakov and Shafarevich, while giving counterexamples to some of their statements. We bound the dimension of the space of global vector fields on an elliptic surface if the generic fiber of is ordinary or if admits no multiple fibers, and show that, without these assumptions, the number can be arbitrarily large for any base curve and any field of positive characteristic. If is not isotrivial, we prove that and give a bound on in terms of the genus of and the multiplicity of multiple fibers of . 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 . Finally, we present additional results on horizontal and vertical group scheme actions on elliptic surfaces which can be applied to determine explicitly in many concrete cases.
Gebhard Martin. "Infinitesimal automorphisms of algebraic varieties and vector fields on elliptic surfaces." Algebra Number Theory 16 (7) 1655 - 1704, 2022. https://doi.org/10.2140/ant.2022.16.1655