Isotrivial elliptic surfaces in positive characteristic

TL;DR

This paper investigates isotrivial elliptic surfaces in positive characteristic, classifies their automorphism groups, and analyzes their geometric invariants using equivariantly normal curves.

Contribution

It provides a classification of such surfaces via a contracted product description and completes the classification of their automorphism groups in all characteristics.

Findings

Computed Betti numbers for these surfaces.

Classified maximal automorphism groups in any characteristic.

Analyzed Picard schemes when G is diagonalizable.

Abstract

We study relatively minimal surfaces equipped with a strongly isotrivial elliptic fibration in positive characteristic by means of the notion of equivariantly normal curves introduced and developed recently by Brion. Such surfaces are isomorphic to a contracted product $E\times^G X$, where $E$ is an elliptic curve, $G$ is a finite subgroup scheme of $E$ and $X$ is a $G$-normal curve. Using this description, we compute their Betti numbers to determine their birational classes. This allow us to complete the classification of maximal automorphism groups of surfaces in any characteristic. When $G$ is diagonalizable, we compute additional invariants to study the structure of their Picard schemes.