Density degree sets and the geometry of curves

Abstract

The arithmetic of an algebraic curve ? over a field ? is captured by it’s scheme-theoretic closed points. Roughly, a closed point of degree ? is a rational point over a degree-? extension of ?. We investigate how Zariski dense degree-? points place restrictions on the geometry of ?. For a curve over a discretely valued Henselian field ? which is a cyclic cover of ℙ1, we describe when the degree-? points are Zariski dense in terms of the ramification of the cyclic cover. For curves over number fields ?, we relate having Zariski dense degree-? points over a finite extension of ? to the Ueno locus of the degree-? effective divisors, and compute the Ueno locus of an arithmetically interesting class of curves.