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Fibrés de Schwarzenberger et coniques de droites sauteuses. (Schwarzenberger’s bundles and conics of jumping lines). (French. English, French summaries) Zbl 0955.14009
Summary: To a stable vector bundle \({\mathcal E}\) of rank 2 on the complex projective plane \(\mathbb{P}_2\), W. Barth [Math. Ann. 226, 125-150 (1977; Zbl 0332.32021)] associates its scheme \(S({\mathcal E})\) of jumping lines in \(\mathbb{P}^\vee_2\). When the first Chern class of \({\mathcal E}\) is even \(S({ \mathcal E})\) is a curve. J. Le Potier [in: Vector bundles in algebraic geometry, Proc. 1993 Durham Symp., Lond. Math. Soc. Lect. Notes Ser. 208, 179-239 (1995; Zbl 0847.14005)] asks which bundles are characterized by their curve of jumping lines. In this paper we prove that if \({\mathcal E}\) is a Schwarzenberger bundle with even first Chern class then \(S({\mathcal E})\) determine \({\mathcal E}\); in this case \(S({\mathcal E})\) is of the form \(nC\) where \(C\) is a smooth conic of \(\mathbb{P}^\vee_2\). We also give a precise study of the zero-scheme of a section of a Schwarzenberger bundle. This study gives a simplification of the well known Poncelet curve theory.

MSC:
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14D06 Fibrations, degenerations in algebraic geometry
32L05 Holomorphic bundles and generalizations
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
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References:
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