Quadratic Euler Characteristic of Geometrically Cyclic Branched Coverings

TL;DR

This paper computes the quadratic Euler characteristic of geometrically cyclic branched coverings using Levine's quadratic Riemann-Hurwitz formula, relating it to Euler classes and applying it to coverings of the projective plane.

Contribution

It introduces a method to compute quadratic Euler characteristics of cyclic branched coverings using quadratic Riemann-Hurwitz and relates these to Euler characteristics of base schemes.

Findings

Derived explicit formulas for quadratic Euler characteristics in cyclic branched coverings.

Connected quadratic Euler characteristics of coverings to those of base schemes in specific cases.

Applied the theory to compute quadratic Euler characteristics of double covers of the projective plane.

Abstract

For an $n$-fold geometrically cyclic branched covering $Y$ of a smooth, projective scheme $X$ branched at a smooth closed subscheme $Z\subset X$ with $n \in k^\times$, we compute the quadratic Euler characteristic of $Y$ in terms of certain Euler classes on $X$ and $Z$ using the quadratic Riemann-Hurwitz formula of Levine. In certain cases with $n$ odd, we relate the quadratic Euler characteristic of $Y$ to the quadratic Euler characteristics of $X$ and $Z$, obtaining similar formulae to the situation in topology. As an application, we compute the quadratic Euler characteristic of geometrically cyclic branched double coverings of $\mathbb{P}^2$.