On the existence of ramified abelian covers

TL;DR

This paper establishes conditions for the existence of ramified abelian covers over normal complete varieties and shows such covers are toric varieties under specific conditions, extending understanding of branched covers in algebraic geometry.

Contribution

It provides explicit criteria for the existence of abelian covers with prescribed ramification data and demonstrates that certain branched covers of toric varieties are themselves toric.

Findings

Conditions for the existence of abelian covers with specified branch data.

Proof that covers of toric varieties are toric under characteristic zero or Galois conditions.

Extension of the theory of branched covers in algebraic geometry.

Abstract

Given a normal complete variety $Y$ over an algebraically closed field $\mathbb K$, distinct effective Weil divisors $D_1,... D_n$ of $Y$ and positive integers $d_1,... d_n$, we spell out the conditions for the existence of an abelian cover of $Y$ branched with order $d_i$ on $D_i$. As an application, we prove that a cover of a normal complete toric variety branched on the torus-invariant divisors is itself a toric variety if the characteristic of $\mathbb K$ is equal to 0 or if the cover is Galois of degree not divisible by the characteristic.