Higher rank Brill--Noether theory on sections of \(K3\) surfaces (Q2909471)

The rank-\(n\) Clifford index of a smooth projective curve is defined with semi-stable vector bundles of rank \(n\) in a similar way as the usual Clifford index in terms of line bundles. Mercat's conjecture \((M_n)\) says that it is equal to the usual Clifford index. It is known that \((M_2)\) holds for various classes of curves. In a recent paper [Pure Appl. Math. Q. 7, No. 4, 1265--1295 (2011; Zbl 1316.14059)], the authors showed that \((M_2)\) is valid for general curves of genus \(\leq 16\) and gave counter-examples for all genera \(\geq 11\), which were provided by a bundle with 4 sections.NEWLINENEWLINE The first result of this paper is that \((M_2)\) fails for any number of sections and for any odd genus \(\geq 11\) along a Noether-Lefschetz divisor inside the locus of curves on \(K3\) surfaces. Moreover, a detailed proof of \((M_2)\) for general curves of genus 11 is given, applying results of Mukai. Finally it is shown that \(M_3\) fails for any genus \(=9\) or \(\geq 11\) generically along the locus of curves lying on a \(K3\) surface.