Galois group of exceptional curves on the generic del Pezzo surface

TL;DR

This paper proves the Galois action on exceptional curves of generic del Pezzo surfaces is maximal across all degrees and fields, and shows most cubic surfaces over certain fields lack Brauer-Manin obstruction.

Contribution

It establishes the maximality of Galois action on exceptional curves for all degrees and fields, and deduces a generic non-obstruction result for cubic surfaces.

Findings

Galois action is maximal on exceptional curves for all degrees d.

Over _q(u), 100% of cubic surfaces have no Brauer-Manin obstruction.

The result holds over any field for the generic del Pezzo surface.

Abstract

We prove that the Galois action on the exceptional curves on the generic del Pezzo surface of degree $d$ is maximal for all degrees $d$ and over any field $k$. As a consequence of the case $d=3$, we deduce that over $\mathbb{F}_q(u)$, 100% of cubic surfaces have no Brauer-Manin obstruction.