Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T06:52:12.709Z Has data issue: false hasContentIssue false
  • English
  • Français

BRILL–NOETHER THEOREMS AND GLOBALLY GENERATED VECTOR BUNDLES ON HIRZEBRUCH SURFACES

Part of: Families, fibrations Surfaces and higher-dimensional varieties (Co)homology theory

Published online by Cambridge University Press:  05 July 2018

IZZET COSKUN  and
JACK HUIZENGA Open the ORCID record for JACK HUIZENGA [Opens in a new window]
IZZET COSKUN
Affiliation:
Department of Mathematics, Statistics and CS, University of Illinois at Chicago, Chicago, IL 60607, USA email coskun@math.uic.edu
JACK HUIZENGA
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA email huizenga@psu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we show that the cohomology of a general stable bundle on a Hirzebruch surface is determined by the Euler characteristic provided that the first Chern class satisfies necessary intersection conditions. More generally, we compute the Betti numbers of a general stable bundle. We also show that a general stable bundle on a Hirzebruch surface has a special resolution generalizing the Gaeta resolution on the projective plane. As a consequence of these results, we classify Chern characters such that the general stable bundle is globally generated.

MSC classification

Primary: 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14J26: Rational and ruled surfaces 14D20: Algebraic moduli problems, moduli of vector bundles 14F05: Sheaves, derived categories of sheaves and related constructions
Type
Article
Information
Nagoya Mathematical Journal , Volume 238 , June 2020 , pp. 1 - 36
Copyright
© 2018 Foundation Nagoya Mathematical Journal

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Izzet Coskun was partially supported by the NSF grant DMS-1500031. Jack Huizenga was partially supported by the NSA Young Investigator Grant H98230-16-1-0306.

References

Beauville, A., Complex Algebraic Surfaces, London Mathematical Society Lecture Note Series 68, Cambridge University Press, Cambridge, 1983.Google Scholar
Bertram, A., Goller, T. and Johnson, D., Le Potier’s strange duality, quot schemes and multiple point formulas for del Pezzo surfaces, preprint, arXiv:1610.04185.Google Scholar
Coskun, I., Degenerations of surface scrolls and the Gromov–Witten invariants of Grassmannians, J. Algebraic Geom. 15 (2006), 223–284.Google Scholar
Coskun, I. and Huizenga, J., “The birational geometry of the moduli spaces of sheaves on ℙ2”, in Proceedings of the Gökova Geometry-Topology Conference 2014, 2015, 114–155.Google Scholar
Coskun, I. and Huizenga, J., Weak Brill–Noether for rational surfaces, Contemp. Math. (to appear).Google Scholar
Eisenbud, D., The Geometry of Syzygies, Springer, 2005.Google Scholar
Fulton, W. and Lazarsfeld, R., Positive Polynomials for Ample Vector Bundles, Ann. of Math. (2) 188(1) (1983), 35–60.Google Scholar
Gaeta, F., Sur la distribution des degrés des formes appartenant à la matrice de l’idéal homogène attaché à un groupe de N points génériques du plan, C. R. Acad. Sci. Paris 233 (1951), 912–913.Google Scholar
Gieseker, D., p-ample bundles and their Chern classes, Nagoya Math. J. 43 (1971), 91–116.10.1017/S0027763000014380Google Scholar
Göttsche, L. and Hirschowitz, A., “Weak Brill–Noether for vector bundles on the projective plane”, in Algebraic Geometry (Catania, 1993/Barcelona), Lecture Notes in Pure and Applied Mathematics 200, Dekker, New York, 1994, 63–74.Google Scholar
Hartshorne, R., Algebraic Geometry, Graduate Texts in Mathematics 52, Springer, New York, 1977.Google Scholar
Hirschowitz, A. and Laszlo, Y., Fibrés génériques sur le plan projectif, Math. Ann. 297(1) (1993), 85–102.10.1007/BF01459489Google Scholar
Huizenga, J., Effective divisors on the Hilbert scheme of points in the plane and interpolation for stable bundles, J. Algebraic Geom. 25(1) (2016), 19–75.Google Scholar
Huybrechts, D. and Lehn, M., The Geometry of Moduli Spaces of Sheaves, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2010.Google Scholar
Lazarsfeld, R., Positivity in Algebraic Geometry II, Springer, 2004.Google Scholar
Le Potier, J., Lectures on Vector Bundles, Cambridge Studies in Advanced Mathematics 54, Cambridge University Press, Cambridge, 1997, translated by A. Maciocia.Google Scholar
Walter, C., “Irreducibility of moduli spaces of vector bundles on birationally ruled surfaces”, in Algebraic Geometry (Catania, 1993/Barcelona), Lecture Notes in Pure and Applied Mathematics 200, 1998, 201–211.Google Scholar