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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
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