Riv. Mat. Univ. Parma, Vol. 6, No. 2, 2015

Edoardo Ballico[1]

Rank \(r\) spanned vector bundles with extremal Chern classes on a smooth surface

Pages: 287-303
Received: 1 January 2015   
Accepted in revised form: 2 October 2015
Mathematics Subject Classification (2010): 14J60.
Keywords:Spanned vector bundles, vector bundles on surfaces, k-spanned line bundle, adjoint line bundle.
Author address:
[1] : Dept. of Mathematics , University of Trento, 38123 Povo (TN), Italy

The author was partially supported by MIUR and GNSAGA of INdAM (Italy).

Abstract: We study spanned vector bundles with extremal Chern classes and large rank on very simple smooth surfaces \(X\) (e.g. on \(\mathbb{P}^2\), following the rank two case solved by Ph. Ellia). Let \(\cal{L}\) be a spanned and ample line bundle on \(X\). Let \(\cal{E}\) be a rank \(r\) spanned vector bundle with \(\det (\cal{E} )\cong \cal{L}\) and no trivial factor. We prove that \(r\le h^0(\cal{L})-1\) and classify all \(\cal{E}\) with \(h^0(\cal{L} )-r-1 \le \alpha\), where \(\alpha\) is the maximal integer \(k\) such that the adjoint line bundle \(\cal{L} \otimes \omega _X\) is \(k\)-spanned.

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