Bounding geometrically integral del Pezzo surfaces

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 $\varepsilon $ -klt del Pezzo surfaces over arbitrary fields of characteristic different from $2, 3$ and $5$ .