Volumes of Zariski chambers (Q1946091)

Let \(X\) be a smooth projective surface defined over the field of complex numbers and let \(P\) be a nef and big divisor on \(X\). Then the Zariski chamber \(Z_{P}\) is defined to be the set of all big divisors whose negative part in the Zariski decomposition has support \(\text{Null}(P)\). Here the set \(\text{Null}(P)\) consists of irreducible curves having intersection zero with \(P\). In previous work [J. Reine Angew. Math. 576, 209--233 (2004; Zbl 1055.14007)] (resp. [J. Algebra 324, No. 1, 92--101 (2010; Zbl 1198.14034)]) Zariski chambers have been studied with respect to geometric (resp. combinatorial) aspects. In this paper, the authors studied the Zariski chambers with a metric point of view, and obtained two main results. In the first one, the authors gave a precise condition for the nef cone volume to be finite, and provided a method for computing it inductively. In the second one, the authors proved that we can determine the volumes of arbitrary Zariski chambers from nef cone volumes of blow-downs. The authors also gave two applications of the second main result. First the authors determined all chamber volumes on Del Pezzo surfaces. As a second application, the authors studied chambers on certain surfaces with big but non-ample anticanonical divisor.