Hilbert-Kunz functions of a Hirzebruch surface

TL;DR

This paper calculates the Hilbert-Kunz functions for all nontrivial ruled surfaces over the projective line, considering all ample line bundles, providing explicit formulas and insights into their algebraic properties.

Contribution

It provides the first comprehensive computation of Hilbert-Kunz functions for ruled surfaces over P^1 with respect to all ample line bundles.

Findings

Explicit formulas for Hilbert-Kunz functions of ruled surfaces

Analysis of how these functions vary with different line bundles

Enhanced understanding of algebraic properties of these surfaces

Abstract

Here we compute Hilbert-Kunz functions of any nontrivial ruled surface over ${\bf P}^1_k$, with respect to all ample line bundles on it.