Arithmetic surjectivity for zero-cycles

Let $f : X \to Y$ be a proper, dominant morphism of smooth varieties over a number field $k$. When is it true that for almost all places $v$ of $k$, the fibre $X_P$ over any point $P \in Y (k_v)$ contains a zero-cycle of degree $1$? We develop a necessary and sufficient condition to answer this question.

The proof extends logarithmic geometry tools that have recently been developed by Denef and Loughran–Skorobogatov–Smeets to deal with analogous Ax–Kochen type statements for rational points.

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This work was supported by the Engineering and Physical Sciences Research Council [EP/ L015234/1], the EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London.

Received 10 December 2018

Accepted 14 October 2019

Published 12 January 2021