Multiplicities associated to graded families of ideals

By Steven Dale Cutkosky. 2012

77 citations recorded by the source corpus.

We prove that limits of multiplicities associated to graded families of ideals exist under very general conditions. Most of our results hold for analytically unramified equicharacteristic local rings, with perfect residue fields. We give a number of applications, including a "volume = multiplicity" formula, generalizing the formula of Lazarsfeld and Mustata, and a proof that the epsilon multiplicity of Ulrich and Validashti exists as a limit for ideals in rather general rings, including analytic local domains. We also prove an asymptotic "additivity formula" for limits of multiplicities, and a formula on limiting growth of valuations, which answers a question posed by the author, Kia Dalili and Olga Kashcheyeva. Our proofs are inspired by a philosophy of Okounkov, for computing limits of multiplicities as the volume of a slice of an appropriate cone generated by a semigroup determined by an appropriate filtration on a family of algebraic objects.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/725f5499fd2529fc5f33db71c370717604f82149

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In Algebra and Number Theory

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Citations in the corpus

Vanishing sequences and Okounkov bodies (2013, background): “For a valuation v centered at a closed point, we prove that v has linear growth iff it has volume zero in the sense of [7,11,19] (cf.”
Kähler–Einstein metrics and volume minimization (2016, background): “By [24], [49] and [18], the limitsup on the right hand side is a limit and is equal to the multiplicity of the graded family of ideals a• = {ap}:”
Thresholds, valuations, and K-stability (2017, background): “18a, Lemma 3.5]. 1.4. Volume. Let vbe a valuation centered at a closed point ˘2X. The volume of vis vol(v) := lim !+1 ‘(O X;˘=a (v)) n=n! 2[0;+1); the existence of the limit being a consequence of [Cut13]. The volume function is homogenous of order n, i.e. vol(tv) = t nvol(v) for t>0. 1.5. Log di”
Asymptotic Multiplicities of Graded Families of Ideals and Linear Series (2014, methodology): “In our paper [6], we prove that such limits exist for graded families of mR-primary ideals, with the restriction that R is an analytically unramified (N = 0) equicharacteristic local ring with perfect residue field (Theorem 5.”
Convex bodies and multiplicities of ideals (2013, methodology): “We are also thankful to Bernard Teissier, Dale Cutkosky, Francois Fillastre and Mircea Mustaţă for informing us about their interesting papers [24, 3, 4, 7, 19].”