Multidegrees, families, and integral dependence
arxiv(2024)
摘要
We study the behavior of multidegrees in families and the existence of
numerical criteria to detect integral dependence. We show that mixed
multiplicities of modules are upper semicontinuous functions when taking fibers
and that projective degrees of rational maps are lower semicontinuous under
specialization. We investigate various aspects of the polar multiplicities and
Segre numbers of an ideal and introduce a new invariant that we call
polar-Segre multiplicities. In terms of polar multiplicities and our new
invariants, we provide a new integral dependence criterion for certain families
of ideals. By giving specific examples, we show that the Segre numbers are the
only invariants among the ones we consider that can detect integral dependence.
Finally, we generalize the result of Gaffney and Gassler regarding the
lexicographic upper semicontinuity of Segre numbers.
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