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

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

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
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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

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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.

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