A Chebotarev-type density theorem for divisors on algebraic varieties

TL;DR

This paper establishes a Chebotarev-type density theorem for divisors on algebraic varieties, linking Galois cover properties to divisor decomposition behavior over fields.

Contribution

It introduces a new density theorem for divisors on varieties and classifies Galois covers based on divisor decomposition patterns.

Findings

Proves a Chebotarev-type density theorem for divisors on varieties.

Classifies Galois covers using divisor decomposition behavior.

Provides a framework for understanding divisor behavior in algebraic geometry.

Abstract

Let $Z \to X$ be a finite branched Galois cover of normal projective geometrically integral varieties of dimension $d \geq 2$ over a perfect field $k$. For such a cover, we prove a Chebotarev-type density result describing the decomposition behaviour of geometrically integral Cartier divisors. As an application, we classify Galois covers among all finite branched covers of a given normal geometrically integral variety $X$ over $k$ by the decomposition behaviour of points of a fixed codimension $r$ with $0 < r < \dim X$.