Geometry and arithmetic of geometrically integral regular del Pezzo surfaces

TL;DR

This paper classifies certain del Pezzo surfaces over imperfect fields and explores their geometric properties, including the existence of sections in fibrations and conditions for rationality of the total space.

Contribution

It provides a classification of geometrically integral regular del Pezzo surfaces over imperfect fields and establishes new results on sections and rationality of del Pezzo fibrations.

Findings

Existence of sections in three-dimensional terminal del Pezzo fibrations over curves.

Rationality of the total space when the base is rational and fibers have anticanonical degree ≥ 5.

Classification of geometrically integral regular del Pezzo surfaces over imperfect fields.

Abstract

We classify geometrically integral regular del Pezzo surfaces which are not geometrically normal over imperfect fields of positive characteristic. Based on this classification, we show that a three-dimensional terminal del Pezzo fibration onto a curve over an algebraically closed field always admits a section. Moreover, we prove that the total space is rational if the base curve is rational and the anticanonical degree of a fibre is at least five.