Persistence of the Brauer–Manin obstruction on cubic surfaces

Let $X$ be a cubic surface over a global field $k$. We prove that a Brauer–Manin obstruction to the existence of $k$-points on $X$ will persist over every extension $L/k$ with degree relatively prime to $3$. In other words, a cubic surface has nonempty Brauer set over $k$ if and only if it has nonempty Brauer set over some extension $L/k$ with $3 \nmid [L:k]$. Therefore, the conjecture of Colliot–Thélène and Sansuc on the sufficiency of the Brauer-Manin obstruction for cubic surfaces implies that $X$ has a $k$-rational point if and only if $X$ has a $0$-cycle of degree $1$. This latter statement is a special case of a conjecture of Cassels and Swinnerton–Dyer.

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During the preparation of this article, the first author was partially supported by NSF DMS-2101434 and the second author was partially supported by NSF DMS-1553459, NSF DMS-2101434, and a Simons Fellowship.

Received 30 November 2021

Received revised 31 March 2022

Accepted 26 April 2022

Published 4 May 2023