Multiplicities of classical varieties

Abstract

The j‐multiplicity plays an important role in the intersection theory of Stückrad–Vogel cycles, while recent developments confirm the connections between the ϵ‐multiplicity and equisingularity theory. In this paper, we establish, under some constraints, a relationship between the j‐multiplicity of an ideal and the degree of its fiber cone. As a consequence, we are able to compute the j‐multiplicity of all the ideals defining rational normal scrolls. By using the standard monomial theory, we can also compute the j‐ and ϵ‐multiplicity of ideals defining determinantal varieties: The found quantities are integrals which, quite surprisingly, are central in random matrix theory.

Mixed multiplicities, Segre numbers and Segre classes

“…As an application of this result we can describe the degree of the fiber cone of an ideal in terms of a classical mixed multiplicity, see Theorem 3.3 and Remark 5. This generalizes a recent result of Jeffries, Montaño and Varbaro [25,Theorem 3.1].…”

“…Then, by the conversion formulas, we obtain the Segre class and the generalized Samuel multiplicities of rational normal scrolls. For the first generalized Samuel multiplicity, the so-called j-multiplicity, there was already a formula by Jeffries, Montaño and Varbaro [25,Theorem 3.3].…”

Blowups and Fibers of Morphisms

“…Moreover, we are able to relate the degree of the map Ψ, the degree of the image, and the j-multiplicity of the ideal I. Results of this type were obtained before by Simis, Ulrich and Vasconcelos [49], Validashti [52], Xie [53] and by Jeffries, Montano, and Varbaro [32].…”

Symbolic Powers of Ideals Defining F-pure and Strongly F-regular Rings

“…Note that we can obtain the same containment results for the ideal of t × t minors of a symmetric n × n matrix or the ideal of 2t-Pfaffians of a generic n × n matrix, using Proposition 4.3 and Theorem 4.4 in [JMV15] for the symmetric case and Theorem 2.1 and Theorem 2.4 in [DN] for the Pfaffians.…”