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arXiv · 1205.3817

Volumes of Zariski chambers

Thomas Bauer·David Schmitz
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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.

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Thomas Bauer, David Schmitz. 2012-05-16. Volumes of Zariski chambers. https://arxiv.org/abs/1205.3817

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