Hilbert-Kunz multiplicities in characteristic zero

This talk concerns the question of whether a simpler limit can be used to obtain the limit Hilbert Kunz multiplicity, a possible candidate for a characteristic zero Hilbert-Kunz multiplicity.
I will first introduce this multiplicity, give some of its history, and finally discuss an affirmative answer for one of the main cases for which it is known to exist, namely that of the affine cones of smooth projective curves. The proof involves more careful estimates of bounds found independently by Brenner and Trivedi on the dimensions of the cohomologies of twists of certain syzygy bundles as the characteristic p goes to infinity and uses asymptotic results of Trivedi on the slopes of Harder-Narasimham filtrations of Frobenius pullbacks of bundles.
This is joint work with Holger Brenner and Jinjia Li.