Isotrivial elliptic surfaces in positive characteristic

TL;DR

The paper studies smooth projective surfaces in positive characteristic equipped with strongly isotrivial elliptic fibrations, reinterpreting them as contracted products $S\cong E\times^G X$ via equivariantly normal curves. It computes Betti numbers, derives when such surfaces have Kodaira dimension $-\infty,0,1$ and classifies cases, including when the acting group $G$ is diagonalizable. The work shows that, in the diagonalizable case, the Picard scheme is reduced and the irregularity and Euler characteristic are determined by the base curve $Y$ and the genus of the intermediate curve $X$, with explicit formulas for $\deg\omega_X$ and the Kodaira dimension. Overall, it extends the birational classification of maximal automorphism groups to positive characteristic and clarifies how diagonalizable quotient data control invariants like Picard ranks and Albanese structures, providing a clear framework for further study of isotrivial elliptic surfaces in this setting.

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.

Theorems & Definitions (104)

• Theorem 1
• Theorem 2
• Corollary 3
• Proposition 4
• Theorem 1.1: Blanchard's Lemma
• Corollary 1.2
• Lemma 1.3
• Lemma 1.4
• proof
• Definition 1.5