## Asymptotic invariants of base loci.(English)Zbl 1127.14010

Let $$D$$ be a big divisor on a normal projective variety $$X$$, and let $${\mathbf B}(D) = \bigcap_{m > 0} \text{ Bs}(mD)$$ denote the stable base locus of $$D$$, where $$\text{ Bs}(mD)$$ is the base locus of the linear system $$| mD|$$. One then defines the augmented base locus $${\mathbf B}_+(D) = \bigcap_A {\mathbf B}(D-A)$$ and the restricted base locus $${\mathbf B}_-(D) = \bigcap_A {\mathbf B}(D+A)$$, where the intersections are taken over all ample $${\mathbb Q}$$-divisors $$A$$. Another asymptotic invariant of divisors introduced in this paper is given by the asymptotic order of vanishing of $$D$$ along a discrete valuation $$v$$ of the function field of $$X$$, which is defined by
$v(\| D \| ) = \lim_{p \to \infty} \frac{v(\text{ Bs}(pD))}{p}.$
These notions are shown to depend only on the numerical class of $$D$$ (contrary to the pathological behavior of stable base loci) and to naturally extend to invariants defined for classes of big $${\mathbb R}$$-divisors. Moreover, the asymptotic order of vanishing extends to a continuous function on the Néron-Severi space of $$X$$. Assuming that $$X$$ is smooth, it is also proven that if $$\xi$$ is the class of a big $${\mathbb R}$$-divisor, then the center $$Z$$ of a discrete valuation $$v$$ is contained in $${\mathbf B}_-(\xi)$$ if and only if $$v(\| \xi\| ) > 0$$. Similar asymptotic invariants are defined starting from other invariants of singularities, such as the Hilbert–Samuel multiplicity and the Arnold multiplicity (which is the inverse of the log canonical threshold). If $$X$$ has finitely generated linear series, the notion being inspired to that of “Mori dream spaces” introduced in Y. Hu and S. Keel [Mich. Math. J. 48, 331–348 (2000; Zbl 1077.14554)], then it is proven that the pseudo-effective cone $$\overline{\text{ Eff}}(X)_{\mathbb R}$$ of $$X$$ is rational polyhedral, and that there is a fan $$\Delta$$ supported precisely on this cone, such that for every $$v$$ the function $$v(\| . \| )$$ is linear on each of the cones in $$\Delta$$.

### MSC:

 14C20 Divisors, linear systems, invertible sheaves 14B05 Singularities in algebraic geometry 14F17 Vanishing theorems in algebraic geometry

### Keywords:

base loci; asymptotic invariants; multiplier ideals

Zbl 1077.14554
Full Text:

### References:

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