Arithmetic behaviour of Frobenius semistability of syzygy bundles for plane trinomial curves

TL;DR

This paper studies how Frobenius semistability of syzygy bundles on plane trinomial curves varies with prime characteristic, revealing a congruence-based behavior and conditions for strong semistability in reductions.

Contribution

It establishes a congruence-dependent pattern for Frobenius semistability of bundles and links characteristic zero semistability to strong semistability in positive characteristic.

Findings

Frobenius semistability depends on p modulo 2λ_h.

Semistability in characteristic zero implies strong semistability for a Zariski dense set of primes.

Existence of a common Zariski dense set of primes for finitely many semistable bundles.

Abstract

Here we consider the set of bundles $\{V_n\}_{n\geq 1}$ associated to the plane trinomial curves $k[x,y,z]/(h)$. We prove that the Frobenius semistability behaviour of the reduction mod $p$ of $V_n$ is a function of the congruence class of $p$ modulo $2\lambda_h$ (an integer invariant associated to $h$). As one of the consequences of this, we prove that if $V_n$ is semistable in characteristic 0, then its reduction mod $p$ is strongly semistable, for $p$ in a Zariski dense set of primes. Moreover, for any given finitely many such semistable bundles $V_n$, there is a common Zariski dense set of such primes.