On multiplicities of graded sequences of ideals
By M. Mustaţă. 2002
96 citations recorded by the source corpus.
We generalize a result of Ein-Lazarsfeld-Smith (math.AG/0202303), proving that for an arbitrary sequence of zero-dimensional ideals, the multiplicity of the sequence is equal with its volume. This is done using a deformation to monomial ideals. As a consequence of our result, we obtain a formula which computes the multiplicity of an ideal I in terms of the multiplicities of the initial monomial ideals of the powers I^m. We use this to give a new proof of the inequality between multiplicity and the log canonical threshold due to de Fernex, Ein and the author.
Subfields: Algebra and Number Theory, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/5994a119120e4f43cababba5ef5201f4b60d54c7
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In Algebra and Number Theory
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Citations in the corpus
- 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}:”
- Uniform approximation of Abhyankar valuation ideals in smooth function fields (2002, unknown): “…he shows that the volume of an arbitrary graded system of ideals is equal to the limit of the normalized multiplicities ^^- in general?that is, that the first equality in the conclusion of Proposition 3.11 holds without the assumption that there exists a 6 such that 8)m c am for all m; see [19].”
- Valuations and asymptotic invariants for sequences of ideals (2010, background): “As is always the case when dealing with graded sequences of ideals (see [34], [15], [37] and [20]), a key tool is provided by the corresponding system of asymptotic multiplier ideals b• = (bt)t∈R>0 .”
- 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].”
- Convex Bodies Associated to Linear Series (2008, background): “Starting (as did Okounkov) with a finite dimensional subspace L ⊆ K(X) of the field of rational functions on a variety X, Kaveh and Khovanskii associate to L a convex body ∆(L) depending on the choice of a valuation ν : K(X)∗ −→ Zd.”
- Existence of valuations with smallest normalized volume (2016, background): “Note that the proof of Proposition 8.3 is directly inspired by the main argument in [Mus02].”
- An algebraic approach to the openness conjecture of Demailly and Kollár (2012, background): “For proofs and details we refer to [19]; see also [26].”