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Zariski chambers, volumes, and stable base loci. (English) Zbl 1055.14007

Let \(X\) be an \(n\)-dimensional smooth projective variety and \(L\) a line bundle on \(X\). Let \(\text{vol} _X(L):= \limsup _k h^0(X,L^{\otimes k})\cdot n!/k!\) be the asymptotic volume of \(L\). This important invariant was introduced by S. Cutkosky [Duke Math. J. 53, 149–156 (1986; Zbl 0604.14002)] and studied by Demailly, Ein, Lazarfeld, Nakamaye and others. It was recently proved [R. Lazarsfeld, “Positivity in algebraic geometry”, Vol. I, Vol. II (2004; Zbl 1066.14021)] that \(\text{vol} _X(L)\) is a continuous log-concave function on the Néron-Severi group. Here the authors prove that if \(n=2\) the cone of big divisors of \(X\) has a locally finite decomposition into locally polyhedral subcones such that the restriction of \(\text{vol}_X(L)\) to each subcone is polynomial, but that this is not true for \(n \geq 3\). They apply their results to the study of the stable loci of \(L\) for varying \(L\) (on the interiors of these subcones the stable base locus is constant). Among the application is a continuity theorem for the Zariski decomposition of \(\mathbb{R}\)-divisors (\(n=2\)).
Editorial remark: See also [the first author and M. Funke, Forum Math. 24, No. 3, 609–625 (2012; Zbl 1242.14007)] for a correction.

MSC:

14C22 Picard groups
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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