Hostname: page-component-7479d7b7d-68ccn Total loading time: 0 Render date: 2024-07-16T01:34:29.087Z Has data issue: false hasContentIssue false

On vector bundles destabilized by Frobenius pull-back

Published online by Cambridge University Press:  17 May 2006

Kirti Joshi
Affiliation:
Department of Mathematics, University of Arizona, Tucson, AZ 85721, USAkirti@math.arizona.edu
S. Ramanan
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, Indiaramanan@math.tifr.res.in
Eugene Z. Xia
Affiliation:
Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan, Republic of Chinaezxia@mail.ncku.edu.tw
Jiu-Kang Yu
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47906, USAjyu@math.purdue.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let X be a smooth projective curve of genus $g > 1$ over an algebraically closed field of positive characteristic. This paper is a study of a natural stratification, defined by the absolute Frobenius morphism of X, on the moduli space of vector bundles. In characteristic two, there is a complete classification of semi-stable bundles of rank 2 which are destabilized by Frobenius pull-back. We also show that these strata are irreducible and obtain their respective dimensions. In particular, the dimension of the locus of bundles of rank two which are destabilized by Frobenius is $3g-4$. These Frobenius destabilized bundles also exist in characteristics two, three and five with ranks 4, 3 and 5, respectively. Finally, there is a connection between (pre)-opers and Frobenius destabilized bundles. This allows an interpretation of some of the above results in terms of pre-opers and provides a mechanism for constructing Frobenius destabilized bundles in large characteristics.

Type
Research Article
Copyright
Foundation Compositio Mathematica 2006