On the geometry of moduli spaces of coherent systems on algebraic curves.
| dc.contributor.author | Bradlow, S.B. | |
| dc.contributor.author | García Prada, O. | |
| dc.contributor.author | Mercat, V. | |
| dc.contributor.author | Muñoz, Vicente | |
| dc.contributor.author | Newstead, P. E. | |
| dc.date.accessioned | 2023-06-20T10:34:23Z | |
| dc.date.available | 2023-06-20T10:34:23Z | |
| dc.date.issued | 2007 | |
| dc.description.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 a. We study the geometry of the moduli space of coherent systems for different values of a when k ≤ n and the variation of the moduli spaces when we vary a. As a consequence, for sufficiently large , we compute the Picard groups and the first and second homotopy groups of the moduli spaces of coherent systems in almost all cases, describe the moduli space for the case k = n − 1 explicitly, and give the Poincare polynomials for the case k = n − 2. In an appendix, we describe the geometry of the “flips” which take place at critical values of a in the simplest case, and include a proof of the existence of universal families of coherent systems when GCD(n, d, k)= 1. | |
| dc.description.department | Depto. de Álgebra, Geometría y Topología | |
| dc.description.faculty | Fac. de Ciencias Matemáticas | |
| dc.description.refereed | TRUE | |
| dc.description.sponsorship | EAGER | |
| dc.description.sponsorship | EDGE | |
| dc.description.sponsorship | European Scientific Exchange Programme (Royal Society of London ) | |
| dc.description.sponsorship | Consejo Superior de Investigaciones Científicas | |
| dc.description.sponsorship | National Science Foundation | |
| dc.description.status | pub | |
| dc.eprint.id | https://eprints.ucm.es/id/eprint/21040 | |
| dc.identifier.doi | 10.1142/S0129167X07004151 | |
| dc.identifier.issn | 0129-167X | |
| dc.identifier.officialurl | http://www.worldscientific.com/doi/abs/10.1142/S0129167X07004151 | |
| dc.identifier.relatedurl | http://www.worldscientific.com | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14352/50593 | |
| dc.issue.number | 4 | |
| dc.journal.title | International journal of mathematics | |
| dc.language.iso | eng | |
| dc.page.final | 453 | |
| dc.page.initial | 411 | |
| dc.publisher | World Scientific | |
| dc.relation.projectID | EC FP5 Contract no. HPRN-CT-2000-00099 | |
| dc.relation.projectID | EC FP5 Contract no. HPRN-CT-2000-00101 | |
| dc.relation.projectID | 15455 and 15646 | |
| dc.relation.projectID | DMS-0072073. | |
| dc.rights.accessRights | open access | |
| dc.subject.cdu | 512.7 | |
| dc.subject.keyword | Algebraic curves | |
| dc.subject.keyword | Moduli of vector bundles | |
| dc.subject.keyword | Coherent systems | |
| dc.subject.keyword | Brill–Noether loci | |
| dc.subject.ucm | Geometria algebraica | |
| dc.subject.unesco | 1201.01 Geometría Algebraica | |
| dc.title | On the geometry of moduli spaces of coherent systems on algebraic curves. | |
| dc.type | journal article | |
| dc.volume.number | 18 | |
| dcterms.references | M. F Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, R. Soc. Lond.Philos. Trans. Ser. A 308 (1982), 523–615. A. Bertram, Stable pairs and stable parabolic pairs, J. Algebraic Geom. 3 (1994), 703–724. S. B. Bradlow and G. Daskalopoulos, Moduli of stable pairs for holomorphic bundles over Riemann surfaces II, Internat. J. Math. 4 (1993), 903–925. S. B. Bradlow and O. Garcıa-Prada, An application of coherent systems to a Brill-Noether problem, J. Reine Angew. Math. 551 (2002), 123–143. L. Brambila-Paz, I. Grzegorczyk and P. E. Newstead,Geography of Brill-Noether loci for small slopes, J. Algebraic Geom. 6 (1997), 645–669. S. B. Bradlow, O. Garcıa-Prada, V. Muñoz and P. E.Newstead, Coherent Systems and Brill-Noether theory, Internat. J. Math. 14 (2003), 683–733. L. Brambila-Paz, V. Mercat, P. E. Newstead and F. Ongay, Nonemptiness of Brill-Noether loci, Internat. J. Math. 11 (2000), 737–760. G. Daskalopoulos, The topology of the space of stable bundles over a compact Riemann surface, J. Differential Geom. 36 (1992), 699–746. G. Daskalopoulos and K. K. Uhlenbeck, An application of transversality to the topology of the moduli space of stable bundles, Topology 34 (1995), 203–215. J.-M. Drezet and M. S. Narasimhan, Groupe de Picard des varietes de modules de fibres semi-stables sur les courbes algebriques, Invent. Math. 97 (1989), 53–94. O. Garcıa-Prada, P. B. Gothen and V. Muñoz, Betti numbers of the moduli space of rank 3 parabolic Higgs bundles,Memoirs Amer. Math. Soc., to appear. P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley & Sons, 1978. A. Grothendieck, Technique de descente et theoremes d’existence en geometrie algebrique. IV Les schemas de Hilbert. S´eminaire Bourbaki 221 (1960/61). M. He, Espaces de modules de syst`emes coh´erents Internat. J. Math. 9 (1998) 545–598. G. Kempf, Toward the inversion of abelian integrals, Ann. of Math. 110 (1979), 243–273. A. King and P. E. Newstead, Moduli of Brill-Noether pairs on algebraic curves, Internat. J. Math. 6 (1995), 733–748. H. Lange and P. E. Newstead, Coherent systems of genus 0, Internat. J. Math. 15 (2004),409–424. H. Lange and P. E. Newstead, Coherent systems on elliptic curves, Internat. J. Math., 16 (2005), 787–805. H. Lange and P. E. Newstead, Coherent systems of genus 0 II: Existence results for k ≥ 3,Internat. J. Math., to appear. J. Le Potier, Faisceaux semi-stables et systemes coherents, Vector Bundles in Algebraic Ge-ometry, Durham 1993, ed. N.J. Hitchin, P.E. Newstead and W.M. Oxbury, LMS Lecture Notes Series 208, 179–239, Cambridge University Press, 1995. A. Mattuck, Picard bundles, Illinois J. Math. 5 (1961), 550–564. A. Mattuck, Symmetric products and Jacobians, Amer. J.Math. 83 (1961), 189–206. I. G. Macdonald, Symmetric products of an algebraic curve, Topology 1 (1961), 319–343. V. Mercat, Le probleme de Brill-Noether pour des fibres stables de petite pente, J. Reine Angew. Math. 506 (1999),1–41. V. Mercat, Le probleme de Brill-Noether et le theoreme de Teixidor, Manuscripta Math. 98 (1999), 75–85. S. Mukai, Duality between D(X) and\D(X) with applications to Picard sheaves, Nagoya Math. J. 81 (1981), 153–175. P. E. Newstead, Topological properties of some spaces of stable bundles, Topology 6 (1967),241–262. P. E. Newstead, Introduction to moduli problems and orbit spaces, T.I.F.R. Lectures on Math.51, Springer-Verlag, 1978. N. Raghavendra and P.A. Vishwanath, Moduli of pairs and generalized theta divisors, Tohoku Math. J. 46 (1994), 321–340. Y. Sepulcre, Factorization of birational morphisms between 4-folds. Ph. Thesis, Paris VII,2004. C. S. Seshadri, Fibres vectoriels sur les courbes algebriques, Asterisque 96 (1982). M. Teixidor i Bigas, Brill-Noether theory for stable vector bundles, Duke Math. J. 62 (1991),123–126. M. Thaddeus, Stable pairs, linear systems and the Verlinde formula, Invent. Math. 117 (1994),317–353. A. N. Tyurin, Classification of vector bundles over an algebraic curve of arbitrary genus, Izv.Akad. Nauk. SSSR Ser. Mat. 29 (1965), 657–688. A. N. Tyurin, Analogues of Torelli’s theorem for multi-dimensional vector bundles on an algebraic curve, Izv. Akad. Nauk SSSR Ser. Math. 34 (1970), 338–365. Translation in Math.USSR-Izv. 4 (1970), 343–370. D. Zagier, Elementary aspects of the Verlinde formula and of the Harder-Narasimhan-Atiyah-Bott formula, Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (RamatGan, 1993) 445–462, Israel Math. Conf. Proc. 9, 1996. | |
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