Volumes of Zariski chambers

TL;DR

This paper introduces a metric perspective on Zariski chambers by defining and computing their volumes, providing new tools for understanding the geometric structure of algebraic surfaces.

Contribution

It establishes conditions for finiteness of nef cone volumes and offers an inductive method to compute Zariski chamber volumes from blow-downs.

Findings

Finite nef cone volume condition established

Method for computing chamber volumes from blow-downs

Explicit calculations on Del Pezzo surfaces

Abstract

Zariski chambers are natural pieces into which the big cone of an algebraic surface decomposes. They have so far been studied both from a geometric and from a combinatorial perspective. In the present paper we complement the picture with a metric point of view by studying a suitable notion of chamber sizes. Our first result gives a precise condition for the nef cone volume to be finite and provides a method for computing it inductively. Our second result determines the volumes of arbitrary Zariski chambers from nef cone volumes of blow-downs. We illustrate the applicability of this method by explicitly determining the chamber volumes on Del Pezzo and other anti-canonical surfaces.