Multiplicities and log canonical threshold

By Tommaso de Fernex, L. Ein, M. Mustaţă. 2002

95 citations recorded by the source corpus.

If R is a local ring of dimension n, of a smooth complex variety, and if I is a zero dimensional ideal in R, then we prove that e(I)\geq n^n/lc(I)^n. Here e(I) is the Samuel multiplicity along I, and lc(I) is the log canonical threshold of (R,I). We show that equality is achieved if and only if the integral closure of I is a power of the maximal ideal. When I is an arbitrary ideal, but n=2, we give a similar bound involving the Segre numbers of I.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1f82ce6dc1e97c48dffd55f577e7895949aa7ac0

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In Geometry and Topology

1. H. Blum
2. M. Mustaţă
3. Chi Li
4. Karl Schwede
5. S. Takagi
6. Kei-ichi Watanabe
7. Craig Huneke
8. Yuri Prokhorov

In Mathematical Physics

1. M. Mustaţă
2. H. Blum
3. S. Takagi
4. Kei-ichi Watanabe
5. J. McKernan
6. Yuri Prokhorov
7. Craig Huneke
8. C. Birkar

Citations in the corpus

• Existence of valuations with smallest normalized volume (2016, methodology): “s, valuations, volumes. This material is based upon work supported by the National Science Foundation under Grant No. DMS-0943832. 1 arXiv:1606.08894v3 [math.AG] 7 Jun 2017 2 HAROLD BLUM Theorem 1.2 ([dFEM04]). Let X be a variety of dimension nand x2X a smooth point. If a O X;x is an ideal that vani”
• IMPANGA lecture notes on log canonical thresholds (2011, methodology): “ariety X. In this case we have the following inequality relating the Hilbert-Samuel multiplicity e(a) of ato the log canonical threshold: (8) e(a) ≥ nn lct(a)n . 12 MIRCEA MUSTAT¸A˘ This is proved in [dFEM3] by first proving a similar inequality for length: (9) ℓ(OX/a) ≥ nn n!lct(a)n . This in turn f”
• F-Thresholds, tight closure, integral closure, and multiplicity bounds (2007, background): “[ dFEM ], which is the special case J = m. However, this version is also open in”
• Threefold thresholds (2003, background): “[22] proceeds by analysing all possible exceptional divisors on a smooth surface that have log discrepancy zero with respect to some divisor KX + ∆.”