The specialization index of a variety over a discretely valued field

TL;DR

This paper introduces the specialization index, a new invariant for proper varieties over henselian discretely valued fields, providing insights into rational points and explaining cases where the index does not suffice.

Contribution

It defines the specialization index, offers an explicit formula via an snc-model, and demonstrates its effectiveness in understanding rational points on varieties.

Findings

Specialization index can differ from the index, explaining absence of rational points.

Explicit formula for the specialization index in terms of an snc-model.

For certain complex varieties with trivial coherent cohomology, the specialization index equals one.

Abstract

Let $X$ be a proper variety over a henselian discretely valued field. An important obstruction to the existence of a rational point on $X$ is the index, the minimal positive degree of a zero cycle on $X$. This paper introduces a new invariant, the specialization index, which is a closer approximation of the existence of a rational point. We provide an explicit formula for the specialization index in terms of an $snc$-model, and we give examples of curves where the index equals one but the specialization index is different from one, and thus explains the absence of a rational point. Our main result states that the specialization index of a smooth, proper, geometrically connected $\mathbb{C}((t))$-variety with trivial coherent cohomology is equal to one.