Classification of rank-2 ample and spanned vector bundles on surfaces whose zero loci consist of general points

Noma, Atsushi

doi:10.1090/S0002-9947-1994-1181186-2

Classification of rank-2 ample and spanned vector bundles on surfaces whose zero loci consist of general points
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Trans. Amer. Math. Soc. 342 (1994), 867-894 Request permission

Abstract:

Let X be an n-dimensional smooth projective variety over an algebraically closed field k of characteristic zero, and E an ample and spanned vector bundle of rank n on X. To study the geometry of (X, E) in view of the zero loci of global sections of E, Ballico introduces a numerical invariant $s(E)$. The purposes of this paper are to give a cohomological interpretation of $s(E)$, and to classify ample and spanned rank-2 bundles E on smooth complex surfaces X with $s(E) = 2{c_2}(E)$, or $2{c_2}(E) - 1$; namely ample and spanned 2-bundles whose zero loci of global sections consist of general ${c_2}(E)$ points or general ${c_2}(E) - 1$ points plus one. As an application of these classification, we classify rank-2 ample and spanned vector bundles E on smooth complex projective surfaces with ${c_2}(E) = 2$.

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