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

2015/11/10 by Benjamin Bakker, Gavril Farkas, Bakker, Benjamin +1 · 1 citation
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG

paper · pdf · doi:10.48550/arxiv.1511.03253

14 pages. Final version, to appear in the American Journal of Mathematics

arxiv created 2016/11/22 · arxiv updated 2016/11/23

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).

Cited by

• Linear stability and rank two Clifford indices of algebraic curves with applications

Related

• Restricted Lazarsfeld-Mukai bundles and canonical curves
• Counterexamples to Mercat's Conjecture
• Sections of K3 surfaces with Picard number two and Mercat's conjecture
• An effective restriction theorem via wall-crossing and Mercat's conjecture
• Higher rank Clifford indices of curves on a K3 surface