N. V. Trung, Institute of Mathematics, Hanoi, Vietnam
Description
Speaker: N. V. Trung, Institute of Mathematics, Hanoi, Vietnam
Date/Time: 29 October 2020, 5:30pm IST/ 12:00 GMT/ 8:00am EDT (joining
time: 5:15 pm IST - 5:30 pm IST)
Google meet link: meet.google.com/zcj-xnpb-ffo
Title: Multiplicity sequence and integral dependence
Abstract: The first numerical criterion for integral dependence was proved
by Rees in 1961 which states that two m-primary ideals $I \subset J$ in an
equidimensional and universally catenary local ring $(R, m)$ have the same
integral closure if and only if they have the same Hilbert-Samuel
multiplicity. This result plays an important role in Teissier's work on
the equisingularity of families of hypersurfaces with isolated
singularities. For hypersurfaces with non-isolated singularities, one
needs a similar numerical criterion for integral dependence of
non-$m$-primary ideals. Since the Hilbert-Samuel multiplicity is no longer
defined for non-$m$-primary ideals, one has to use other notions of
multiplicities that can be used to check for integral dependence. A
possibility is the multiplicity sequence which was introduced by Achilles
and Manaresi in 1997 and has its origin in the intersection numbers of the
Stuckrad-Vogel algorithm. It was conjectured that two arbitrary ideals $I
\subset J$ in an equidimensional and universally catenary local ring have
the same integral closure if and only if they have the same multiplicity
sequence. This talk will present a recent solution of this conjecture by
Polini, Trung, Ulrich and Validashti.