Finite Morphisms to Projective Space and Capacity Theory

TL;DR

This paper characterizes rings allowing finite morphisms from projective schemes to projective space and introduces a new capacity concept for adelic subsets, extending classical theorems in arithmetic geometry.

Contribution

It establishes equivalent conditions on rings for the existence of finite morphisms to projective space and defines a novel capacity for adelic subsets, generalizing Fekete-Szeg ext{"o} theorem.

Findings

Characterization of rings with finite morphisms to projective space

Definition of a new capacity for adelic subsets

Extension of the Fekete-Szeg ext{"o} Theorem

Abstract

We study conditions on a commutative ring R which are equivalent to the following requirement; whenever X is a projective scheme over S = Spec(R) of fiber dimension \leq d for some integer d \geq 0, there is a finite morphism from X to P^d_S over S such that the pullbacks of coordinate hyperplanes give prescribed subschemes of X provided these subschemes satisfy certain natural conditions. We use our results to define a new kind of capacity for adelic subsets of projective schemes X over global fields. This capacity can be used to generalize the converse part of the Fekete-Szeg\H{o} Theorem.