# Restricted Lazarsfeld-Mukai bundles and canonical curves

@article{Aprodu2014RestrictedLB,
title={Restricted Lazarsfeld-Mukai bundles and canonical curves},
author={Marian Aprodu and Gavril Farkas and Angela Ortega},
journal={arXiv: Algebraic Geometry},
year={2014}
}
• Published 2014
• Mathematics
• arXiv: Algebraic Geometry
We prove two results. First, we establish that the normal bundle of any smooth curve of genus 7 having maximal Clifford index is stable. Note that 7 is the smallest genus for which such a result could possibly hold. We then show that rank four Lazarsfeld-Mukai vector bundles on a curve that lies on a general K3 surface are stable. Both results have consequences for Mercat's conjecture on higher rank vector bundles on generic curves.
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#### References

SHOWING 1-10 OF 32 REFERENCES
Lazarsfeld–Mukai Bundles and Applications
We survey the development of the notion of Lazarsfeld-Mukai bundles together with various applications, from the classification of Mukai manifolds to Brill-Noether theory and syzygies of K3 sections.Expand
Clifford Indices for Vector Bundles on Curves
• Mathematics
• 2010
For smooth projective curves of genus g ≥ 4, the Clifford index is an important invariant which provides a bound for the dimension of the space of sections of a line bundle. This is the first step inExpand
CLIFFORD'S THEOREM AND HIGHER RANK VECTOR BUNDLES
We give here a refinement of the classical Clifford's theorem for the upper bound of the number of independent global sections of a semistable vector bundle on a smooth curve. We also conjecture aExpand
Stability of rank-3 Lazarsfeld-Mukai bundles on K3 surfaces
Given an ample line bundle L on a K3 surface S, we study the slope stability with respect to L of rank-3 Lazarsfeld-Mukai bundles associated with complete, base point free nets of type g^2_d onExpand
The Maximal Rank Conjecture and Rank Two Brill-Noether Theory
• Mathematics
• 2010
We describe applications of Koszul cohomology to the BrillNoether theory of rank 2 vector bundles. Among other things, we show that in every genus g > 10, there exist curves invalidating Mercat’sExpand
Bundles of rank 3 on curves of Clifford index 3
• Mathematics, Computer Science
• J. Symb. Comput.
• 2013
This work extends results on one of the Clifford index for vector bundles on a smooth projective curve C of genus g>=4 to the case where C has classical Clifford index 3. Expand
ON AN EXAMPLE OF MUKAI
• Mathematics
• Glasgow Mathematical Journal
• 2011
Abstract In this paper we use an example of Mukai to construct semistable bundles of rank 3 with six independent sections on a general curve of genus 9 or 11 with Clifford index strictly less thanExpand
Green’s conjecture for curves on arbitrary K3 surfaces
• Mathematics
• Compositio Mathematica
• 2011
Abstract Green’s conjecture predicts than one can read off special linear series on an algebraic curve, by looking at the syzygies of its canonical embedding. We extend Voisin’s results on syzygiesExpand
Higher rank Brill-Noether theory on sections of K3 surfaces
• Mathematics
• 2011
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 forExpand
Brill-Noether-Petri without degenerations
The purpose of this note is to show that curves generating the Picard group of a K3 surface X with Pic( X) = Z behave generically from the point of view of Brill-Noether theory. In particular, oneExpand