Finite morphisms from curves over Dedekind rings to $P^1$

TL;DR

This paper provides a new proof that normal projective curves over Dedekind rings with certain fraction fields admit finite morphisms to the projective line, using intersection theory and work of Moret-Bailly.

Contribution

It offers an alternative proof of Green's theorem on finite morphisms from curves over Dedekind rings to P^1, employing intersection theory and existing results.

Findings

Established a new proof of Green's theorem

Demonstrated the application of intersection theory in this context

Extended understanding of morphisms from curves over Dedekind rings

Abstract

A theorem of B. Green states that if A is a Dedekind ring whose fraction field is a local or global field, every normal projective curve over Spec(A) has a finite morphism to P^1_A. We give a different proof of a variant of this result using intersection theory and work of Moret-Bailly.