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Simple vector bundles on plane degenerations of an elliptic curve

Authors: Lesya Bodnarchuk, Yuriy Drozd and Gert-Martin Greuel
Journal: Trans. Amer. Math. Soc. 364 (2012), 137-174
MSC (2000): Primary 16G60; Secondary 14H10, 14H60
Published electronically: August 25, 2011
MathSciNet review: 2833580
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Abstract: In 1957 Atiyah classified simple and indecomposable vector bundles on an elliptic curve. In this article we generalize his classification by describing the simple vector bundles on all reduced plane cubic curves. Our main result states that a simple vector bundle on such a curve is completely determined by its rank, multidegree and determinant. Our approach, based on the representation theory of boxes, also yields an explicit description of the corresponding universal families of simple vector bundles.

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Additional Information

Lesya Bodnarchuk
Affiliation: Max-Planck-Institut für Mathematik, Bonn, Germany

Yuriy Drozd
Affiliation: Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev, Ukraine

Gert-Martin Greuel
Affiliation: Fachbereich Mathematik, University of Kaiserslautern, Kaiserslautern, Germany

Keywords: Simple vector bundles and their moduli, degeneration of an elliptic curve, tame and wild, small reduction.
Received by editor(s): July 28, 2009
Received by editor(s) in revised form: March 3, 2010
Published electronically: August 25, 2011
Additional Notes: We express our sincere thanks to Professor Serge Ovsienko for fruitful discussions and helpful advice. The first named author would like to thank Institut des Hautes Études Scientifiques and the Mathematisches Forschungsinstitut Oberwolfach, where she stayed during the period that this paper was written.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.