The Mercat Conjecture for stable rank 2 vector bundles on generic curves

TL;DR

This paper proves Mercat's conjecture relating the second Clifford index to the classical one for generic curves of any genus, using K3 surface moduli, and identifies a divisor where the conjecture fails.

Contribution

It establishes Mercat's conjecture for generic curves of all genera and characterizes a divisor in the moduli space where the conjecture does not hold.

Findings

Mercat's conjecture holds for generic curves of every genus.

An effective divisor in the moduli space is identified where the conjecture fails.

The slope of this divisor is computed as 6+12/(g+1).

Abstract

It has been a long-standing problem to find an adequate definition of a Clifford index for higher rank vector bundles on curves, which should capture the complexity of the curve in its moduli space. An interesting proposal in rank 2 has been put forward by Mercat, who conjectured that the second Clifford index of a curve should be equal to its classical Clifford index, defined in terms of gonality. Using moduli of sheaves on generic K3 surfaces, we prove Mercat's conjecture for generic curves of every genus. Furthermore, for odd g we identify an effective divisor in the moduli space M_g along which the Mercat Conjecture fails and we compute its slope, which is shown to be equal to 6+12/(g+1).