On the Brauer group of a generic Godeaux surface

TL;DR

This paper investigates the Brauer group of a generic Godeaux surface, showing that the pullback map is injective under certain conditions, using a degeneration technique applicable to similar cases.

Contribution

It demonstrates the injectivity of the Brauer group pullback for Godeaux surfaces with specific Picard rank, introducing a degeneration method for broader applications.

Findings

Injectivity of the Brauer group pullback when $ ho(Y)=9$

Degeneration technique applicable to other algebraic surfaces

Provides new insights into the structure of Godeaux surfaces

Abstract

Let $X$ be a Godeaux surface over $\mathbb{C}$ and $q_{X}\colon Y\to X$ be its universal cover. We show that the pullback map $q^{*}_{X}\colon Br(X)\to Br(Y)$ is injective if $\rho(Y)=9$. Our arguments rely on a degeneration technique that also applies to other examples.