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
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.
[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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