Restricted volumes and base loci of linear series

By L. Ein, R. Lazarsfeld, M. Mustaţă, M. Nakamaye, Mihnea Cristian Popa. 2006

193 citations recorded by the source corpus.

We introduce and study the restricted volume of a divisor along a subvariety. Our main result is a description of the irreducible components of the augmented base locus by the vanishing of the restricted volume.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6ac0a1d2527c4d3e5bafc7434e26ef0b1bf183eb

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Comparing numerical dimensions (2011, methodology): “In fact, we have the following: Proposition 4.12 [Ein et al. 2009, Proposition 2.11, Theorem 2.13].”
Kähler currents and null loci (2013, background): “2 ([23,33]) Let X be an n-dimensional smooth projective variety over C and D a nef R-divisor on X with Dn > 0.”
Convex Bodies Associated to Linear Series (2008, background): “…Wm = H 0 ( X,OX(mD) ) is the complete linear series of a big divisor D, this implies a basic theorem of Fujita ([19], [14], [31], [38], [28, Chapter 11.4]) to the effect that the volume of D can be approximated arbitrarily closely by the self-intersection of an ample divisor on a modification of X.”
Differentiability of volumes of divisors and a problem of Teissier (2006, methodology): “We thus recover with a diﬀerent proof a special case of the main result of [ELMNP3].”
Vanishing sequences and Okounkov bodies (2013, methodology): “When v is divisorial, we can even provide an explicit formula for the density of μL ,v in terms of restricted volumes [10], using the differentiability property of the volume function proved in [4,19].”
Bergman kernels and equilibrium measures for line bundles over projective manifolds (2007, background): “metrized versions of the very recent result in [21] concerning a generalized Fujita approximation theorem for the restricted volume (i.”
Positivity of twisted relative pluricanonical bundles and their direct images (2014, background): “By the main result of [17], it is enough to prove that we have”
K-stability of Fano varieties via admissible flags (2020, unknown): “In the setup of Lemma 4.5, assume that L is ample, and let 𝜋 : 𝑇 → 𝑆 be the prime blowup associated to E.”
The augmented base locus of real divisors over arbitrary fields (2013, methodology): “nts of the base locus which capture more essential properties of divisors and linear systems. These are closely related to important concepts and problems in birational geometry, eg see [4][16][15][9][8][6][2]. We start with some deﬁnitions. The augmented base locus. Let X be a scheme. An R-Cartier ”