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Logarithmic vector bundles on the projective plan. (Fibrés logarithmiques sur le plan projectif.) (French. English summary) Zbl 1139.14019

The author describes the scheme of jumping lines of a logarithmic bundle on the projective plane, with odd first Chern class. In this way he completes a result already known for logarithmic bundles of the projective plane with even first Chern class [I. Dolgachev, M. Kapranov, Duke Math. J. 71, 633–664, (1993; Zbl 0804.14007)]. For the ‘odd’ case, he obtains a description of this scheme adding a finite set of lines to the ones of the divisor of the bundle; this set of lines comes from a connection between the logarithmic bundle and the associated ideal sheaf of the group of points in the projective dual space. The author also gives some explicit constructions which enlighten the geometric meaning of the main result of the paper.

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli

Citations:

Zbl 0804.14007
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References:

[1] Deligne (P.).— Théorie de Hodge II, Publ.Math. IHES, 40, 5-58 (1971). · Zbl 0219.14007
[2] Dolgachev ( I.), Kapranov (M.).— Arrangements of hyperplanes and vector bundles on \({\bf P}_n\), Duke Math.J. 71, 633-664 (1993). · Zbl 0804.14007
[3] Gruson (L.), Peskine (C.).— Courbes de l’espace projectif : variétés de sécantes, Progress in Math 24 (1982). · Zbl 0531.14020
[4] Vallès (J.).— Conique de droites de saut et Fibrés de Schwarzenberger, BSMF, 128, 433-449 (2000). · Zbl 0955.14009
[5] Vallès (J.).— Nombre maximal d’hyperplans instables pour un fibré de Steiner, Math. Zeit., 233, 507-514 (2000). · Zbl 0952.14011
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