Algebra Seminar: Multiplicity Interpolation and a Generalization of a Conjecture of Watanabe and Yoshida
William Taylor, Department of Mathematics, University of Arkansas
Multiplicity Interpolation and a Generalization of a Conjecture of Watanabe and Yoshida
In the first part of the talk we will develop a family of multiplicities, parameterized by a real number s, that interpolates continuously between Hilbert-Kunz and Hilbert-Samuel multiplicities of two ideals of maximal height in a local ring of positive characteristic. We will examine the properties of these s-multiplicities and realize some results on Hilbert-Samuel and Hilbert-Kunz multiplicities as special cases of a more general theory. We will discuss how this unifying framework might give some insight into solving some old conjectures and generalizing the notions of multiplicity to new situations.
In the second part of the talk we will discuss recent joint work with Lance E. Miller, in which we established some bounds on the values of the s-multiplicities. We will show how we generalize some theorems and conjectures about Hilbert-Kunz multiplicity to the s-multiplicity case. In particular, we will show that the s-multiplicity version of the Watanabe-Yoshida minimality conjecture holds in small dimensions.