Multiplicities and log canonical threshold

2002/05/15 by Tommaso de Fernex, Lawrence Ein, Mircea Mustata · 1 citation
Mathematics · #math.AG #msc:14B05 #msc:14C17

paper · pdf

published as J. Alg. Geom. 13 (2004), 603-615. · 13 pages; AMS-LaTeX

arxiv created 2002/05/15 · arxiv updated 2009/11/30

Abstract

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)≥ nⁿ/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.

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