Moduli Spaces of Coherent Systems of Small Slope on Algebraic Curves

Abstract

Let C be an algebraic curve of genus g ≥ 2. A coherent system on C consists of a pair (E, V), where E is an algebraic vector bundle over C of rank n and degree d and V is a subspace of dimension k of the space of sections of E. The stability of the coherent system depends on a parameter α. We study the geometry of the moduli space of coherent systems for 0 < d ≤ 2n. We show that these spaces are irreducible whenever they are nonempty and obtain necessary and sufficient conditions for nonemptiness.

Key Words:

• Algebraic curves
• Brill–Noether loci
• Coherent systems
• Moduli of vector bundles

2000 Mathematics Subject Classification:

• 14H60
• 14D20
• 14H51

ACKNOWLEDGMENTS

The fourth author wishes to acknowledge Universidad Complutense de Madrid and the Institute for Advanced Study, Princeton for their hospitality and for providing excellent working conditions. The fifth author would like to thank CIMAT, Guanajuato, Mexico, and California State University Channel Islands, where parts of this research were carried out.

All authors are members of the research group VBAC (Vector Bundles on Algebraic Curves). Support was received from a grant from the European Scientific Exchange Programme of the Royal Society of London and the Consejo Superior de Investigaciones Científicas (15646) and a further grant from the Royal Society of London for an International Joint Project (2005/R3). The first author was partially supported by the National Science Foundation under grant DMS-0072073. The first, second, and fourth authors were supported through MEC grant MTM2004-07090-C03-01 (Spain) and the fourth also through NSF grant DMS-0111298 (U.S.) during a visit to IAS, Princeton, U.S. The fifth author was supported by the Academia Mexicana de Ciencias during a visit to CIMAT, Guanajuato, Mexico.

Notes

Communicated by L. Ein.