# Vanishing of the Brauer group of a del Pezzo surface of degree 4

## TL;DR

This paper constructs a specific degree 4 del Pezzo surface over a field where the Brauer group behaves unexpectedly, showing limitations of existing computational algorithms.

## Contribution

It provides an explicit example of a del Pezzo surface with a trivial Brauer group despite nontrivial Galois cohomology, highlighting the boundaries of current methods.

## Key findings

- The constructed surface has trivial Brauer group over the base field.
- The Galois cohomology group is nontrivial, isomorphic to Z/2Z.
- The example demonstrates the algorithm's limitations in certain cases.

## Abstract

We explicitly construct a del Pezzo surface $X$ of degree 4 over a field $k$ such that $\operatorname{H}^1(k,\operatorname{Pic}\overline X)$ is isomorphic to $\mathbb{ZZ}/2\mathbb{Z}$ while $\operatorname{Br} X/\operatorname{Br} k$ is trivial. This proves that the algorithm to compute the Brauer group in [VAV] cannot be generalized in some cases.