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Maximal number of unstable hyperplanes for a Steiner bundle. (Nombre maximal d’hyperplans instables pour un fibré de Steiner.) (French) Zbl 0952.14011
Let $${\mathcal S}_{n,k}$$ denote the family of Steiner’s bundle $$S$$ on $$\mathbb{P}_n$$ are defined by an exact sequence $$(k>0)$$ $0\to k{\mathcal O}_{\mathbb{P}_n} (-1)\to (n+k){\mathcal O}_{\mathbb{P}_n}\to S\to 0.$ We show the following result: Let $$S\in{\mathcal S}_{n,k}$$ and $$H_1,\dots, H_{n+k+2}$$ be distincts hyperplanes such that $$h^0 (S_{H_i}^\vee)\neq 0$$. Then it exists a rational normal curve $$C_n \subset \mathbb{P}_n^\vee$$ such that $$H_i\in C_n$$ for $$i= 1,\dots, n+k+ 2$$ and $$S\simeq E_{n+k-1} (C_n)$$, where $$E_{n+k-1} (C_n)$$ is the Schwarzenberger bundle on $$\mathbb{P}_n$$ which belongs to $${\mathcal S}_{n,k}$$ associated to $$C_n \subset \mathbb{P}_n^\vee$$. This implies that a Steiner bundle $$S\in{\mathcal S}_{n,k}$$, if it is not a Schwarzenberger bundle, then it possesses no more than $$(n+k+1)$$ unstable hyperplanes; this proves in any case a result of I. Dolgachev and M. Kapranov [Duke Math. J. 71, No. 3, 633-664 (1993; Zbl 0804.14007), theorem 7.2] about logarithmic bundles.

##### MSC:
 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14J70 Hypersurfaces and algebraic geometry
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