 
Summary: Rend. Sem. Mat. Univ. Pol. Torino  Vol. 64, 4 (2006)
Syzygy 2005
R. Achilles M. Manaresi
GENERALIZED SAMUEL MULTIPLICITIES AND
APPLICATIONS
Dedicated to Paolo Valabrega on the occasion of his 60 th birthday
Abstract. In this note we survey and discuss the main results on the multiplicity sequence we
introduced in former papers as a generalization of Samuel's multiplicity. We relate this new
multiplicity to other numbers introduced in different contexts, for example the Segre numbers
of Gaffney and Gassler and the Hilbert coefficients defined by Ciuperca. Discussing some
examples we underline the usefulness of the multiplicity sequence for concrete calculations
in algebraic geometry using computer algebra systems.
1. Introduction
Intersection theory and singularity theory stimulated the development of a theory of
multiplicities in local rings.
For example, the Samuel multiplicity of the maximal ideal m of a local ring
(A, m) measures the singularity of a variety at some point or along some subvarieties,
and the Samuel multiplicity of an mprimary ideal was introduced in order to define the
intersection number of an irreducible component of the intersection of two varieties X
and Y.
