English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We give several results concerning the connected component ${\rm Aut}_X^0$ ofthe automorphism scheme of a proper variety $X$ over a field, such as itsbehaviour with respect to birational modifications, normalization, restrictionsto closed subschemes and deformations. Then, we apply our results to study theautomorphism 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 thedimension $h^0(X,T_X)$ of the space of global vector fields on an ellipticsurface $X$ if the generic fiber of $f$ is ordinary or if $f$ admits nomultiple 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 ofpositive 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 themultiplicity of multiple fibers of $f$. As a corollary, we re-prove thenon-existence of global vector fields on K3 surfaces and calculate theconnected component of the automorphism scheme of a generic supersingularEnriques surface in characteristic $2$. Finally, we present additional resultson horizontal and vertical group scheme actions on elliptic surfaces which canbe applied to determine ${\rm Aut}_X^0$ explicitly in many concrete cases.

Infinitesimal automorphisms of algebraic varieties and vector fields on elliptic surfaces