## 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)

### Citations:

Zbl 1066.14021; Zbl 0604.14002; Zbl 1242.14007
Full Text:

### References:

 [1] Nakamaye, Stable base loci of linear series, Math Ann pp 318– (2000) · Zbl 1063.14008 [2] Looijenga, Invariant theory of generalized root systems Invent, Math pp 61– (1980) · Zbl 0436.17005 [3] Nakamaye, Base loci of linear series are numerically determined, Trans Amer Math Soc pp 355– (2003) · Zbl 1017.14017 [4] Cossac, Enriques surfaces I Progr Birkha user, Math pp 76– (1989) [5] Kova, cs The cone of curves of a surface no, Math Ann pp 300– (1994) [6] Demazure, Se minaire sur le singularite s des surfaces Lect Springer - Verlag, Notes Math pp 777– (1980) [7] Cutkosky, Zariski decomposition of divisors on algebraic varieties Duke no, Math J pp 53– (1986) · Zbl 0604.14002 [8] Cutkosky, Periodicity of the fixed locus of multiples of a divisor on a surface Duke Math, J pp 72– (1993) · Zbl 0803.14002 [9] Kawamata, Introduction to the minimal model problem Algebraic geometry Stud Pure North Holland Amsterdam, Math pp 283– (1985) [10] Campana, Algebraicity of the ample cone of projective varieties reine, angew Math pp 407– (1990) · Zbl 0728.14004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.