Algebraic and Arithmetic Geometry Seminar

Hilbert-Kunz density function and its applications to some Hilbert-Kunz multiplicity conjectures

by Vijaylaxmi Trivedi (Tata Institute of Fundamental Research, India)

Aula Mancini (Scuola Normale Superiore, Pisa)

Aula Mancini

Scuola Normale Superiore, Pisa


In this talk we introduce a compactly supported and real valued continuous function called Hilbert-Kunz (HK) density function. We briefly describe its properties and its applications to study characteristic p-invariants like HK multiplicity and F -thresholds.
Further we discuss in more detail some long standing conjectures of Watanabe-Yoshida and Yoshida on the HK multiplicities of quadric hypersurfaces.
Here, using the classification of ACM bundles on the smooth quadric via matrix factorizations, we describe the HK density functions of the quadrics. The structure of the function explains the difficulties in nailing down the computations of HK multiplicities of even such simple class of rings.
As a corollary we prove a part of the Watanabe-Yoshida conjecture for all dimensions. Moreover, for large p, we give a closed formula for HK multiplicities of quadrics hypersurfaces and a proof of Yoshida’s conjecture.