# Arithmetic Surjectivity for Zero-Cycles

## TL;DR

This paper establishes a necessary and sufficient condition for when fibers over local points contain zero-cycles of degree one, extending logarithmic geometry techniques to analyze arithmetic properties of morphisms over number fields.

## Contribution

It introduces a new criterion for zero-cycle surjectivity in fibers of morphisms over number fields, expanding the application of logarithmic geometry tools.

## Key findings

- Provides a complete characterization of zero-cycle existence in fibers over local fields.
- Extends recent logarithmic geometry methods to broader arithmetic questions.
- Offers a framework for analyzing Ax-Kochen type statements for zero-cycles.

## Abstract

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.