Infinitesimal automorphisms of algebraic varieties and vector fields on elliptic surfaces

We give several results concerning the connected component ${\rm Aut}_X^0$ of the automorphism scheme of a proper variety $X$ over a field, such as its behaviour 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 $f: X \to C$ over algebraically closed fields, generalizing work of Rudakov and Shafarevich, while giving counterexamples to some of their statements. We bound the dimension $h^0(X,T_X)$ 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 $h^0(X,T_X)$ can be arbitrarily large for any base curve $C$ and any field of positive characteristic. If $f$ is not isotrivial, we prove that ${\rm Aut}_X^0 \cong \mu_{p^n}$ 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 re-prove the non-existence 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 ${\rm Aut}_X^0$ explicitly in many concrete cases.