Degrees of points on varieties over Henselian 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 [Invent. Math. 192 (2013), pp. 567–626] 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 $6$.