On the standard models of del Pezzo fibrations of degree four

TL;DR

This paper proves the existence of standard models for degree 4 del Pezzo fibrations in characteristic greater than 2, extending previous complex case results using Kollár stability techniques.

Contribution

It establishes the existence of standard models of degree 4 del Pezzo fibrations in positive characteristic, a significant extension of prior complex case work.

Findings

Existence of standard models in characteristic > 2

Application of Kollár stability in positive characteristic

Extension of complex case results to new characteristic setting

Abstract

Corti defined the notion of standard models of del Pezzo fibrations, and studied their existence over $\mathbb{C}$ with a fixed generic fibre. In this paper, we prove the existence of standard models of del Pezzo fibrations of degree $4$ in characteristic $>2$. To show this, we use the notion of Koll\'ar stability, which was introduced by Koll\'ar and Abban-Fedorchuk-Krylov.