Degrees of points on varieties over Henselian fields

TL;DR

This paper demonstrates how to compute the entire set of degrees of closed points on varieties over Henselian fields using data from the special fiber, leading to an algorithm for smooth curves and revealing differences from finitely generated fields.

Contribution

It extends previous results by showing the full degree set can be derived from the special fiber, not just the GCD, and provides an explicit algorithm for curves over Henselian fields.

Findings

Degree sets can be computed from special fiber data.

Algorithms are developed for smooth curves over Henselian fields.

Degree sets over such fields can differ significantly from those over finitely generated fields.

Abstract

Let $W/K$ be a nonempty scheme over the field of fractions of a Henselian local ring $R$. A result of Gabber, Liu and Lorenzini shows that the GCD of the set of degrees of closed points on $W$ (which is called the index of $W/K$) can be computed from data pertaining only to the special fiber of a proper regular model of $W$ over $R$. We show that the entire set of degrees of closed points on $W$ can be computed from data pertaining only to the special fiber, provided the special fiber is a strict normal crossings divisor. As a consequence we obtain an algorithm to compute the degree set of any smooth curve over a Henselian field with finite or algebraically closed residue field. Using this we show that degree sets of curves over such fields can be dramatically different than degree sets of curves over finitely generated fields. For example, while the degree set of a curve over a finitely generated field contains all sufficiently large multiples of the index, there are curves over $p$-adic fields with index $1$ whose degree set excludes all integers that are coprime to $6$.