Butler's conjecture for K3 curves

Abstract

Butler's conjectures predict that, on a general smooth curve $C$ of genus $g\ge3$, a general coherent system $[\mathcal{O}_C\otimes V\xrightarrowφE]$ with semistable underlying bundle $E$ and $\dim V>\operatorname{rk}(E)$ has surjective evaluation map $φ$ and stable kernel. In this paper, we prove Butler's conjectures for $g\ge8$ in the two cases in which the Brill-Noether geometry of a general curve is known, namely rank one and the range $μ(E)>2g-2$. By deformation and openness, it suffices to establish the result on a single Brill-Noether-Petri general curve. We do so for curves on K3 surfaces of Picard rank one, using the wall-crossing theory of coherent systems developed by the first two authors.