Higher rank Brill-Noether theory on sections of K3 surfaces

TL;DR

This paper investigates the failure of Mercat's conjecture in higher rank Brill-Noether theory on K3 surfaces, providing counterexamples and confirming the conjecture in specific cases, along with detailed geometric analysis.

Contribution

It demonstrates the failure of Mercat's conjecture in ranks 2 and 3 on K3 surfaces and proves the conjecture for general genus 11 curves, also analyzing Fourier-Mukai involutions.

Findings

Mercat's conjecture in rank 2 fails for any number of sections on K3 surfaces.

Mercat's conjecture in rank 3 fails even for curves with Picard number 1.

The conjecture is confirmed for general genus 11 curves.

Abstract

We discuss the role of K3 surfaces in the context of Mercat's conjecture in higher rank Brill-Noether theory. Using liftings of Koszul classes, we show that Mercat's conjecture in rank 2 fails for any number of sections and for any gonality stratum along a Noether-Lefschetz divisor inside the locus of curves lying on K3 surfaces. Then we show that Mercat's conjecture in rank 3 fails even for curves lying on K3 surfaces with Picard number 1. Finally, we provide a detailed proof of Mercat's conjecture in rank 2 for general curves of genus 11, and describe explicitly the action of the Fourier-Mukai involution on the moduli space M_{11}.