Bounding geometrically integral del Pezzo surfaces

Abstract

We prove several boundedness statements for geometrically integral normal del Pezzo surfaces $X$ over arbitrary fields. We give an explicit sharp bound on the irregularity if $X$ is canonical or regular. In particular, we show that wild canonical del Pezzo surfaces exist only in characteristic 2. As an application, we deduce that canonical del Pezzo surfaces form a bounded family over $\mathbb{Z}$, generalising work of Tanaka. More generally, we prove the BAB conjecture on the boundedness of $\epsilon$-klt del Pezzo surfaces over arbitrary fields of characteristic different from 2, 3, and 5.

26 pages, v2: minor changes. To appear in Forum Math Sigma