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On vector bundles on 3-folds with sectional genus 1. (English) Zbl 0729.14030

If X is a smooth complete connected scheme of dimension n over an algebraically closed field of characteristic zero and E a rank-(n-1) vector bundle on X, the sectional genus g(E) is defined by the formula \(2g(E)-2=(K_ X+c_ 1(E)).c_{n-1}(E)\). This is an integer. If E has a section with a curve C as zero-locus, g(E) is the arithmetical genus of C. Wisniewski studied in his Ph.D. thesis the case \(n=3\), \(g(E)=0\). - The aim of the present paper is to classify the pairs (X,E) where E is a spanned ample vector bundle on X in the case \(n=3\), \(g(E)=1\). The author almost gives a classification in this case. It only remains to describe all such pairs (X,E) where X is a \({\mathbb{P}}_ 2\)-bundle over a smooth elliptic curve C and the restriction of E to every fibre is the direct sum of two line bundles of degree 1. The proof uses Mori theory and the classification of Fano 3-folds.

MSC:

14J30 \(3\)-folds
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J45 Fano varieties
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
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