Maximal number of unstable hyperplanes for a Steiner bundle (Q1974149)

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 \textit{I. Dolgachev} and \textit{M. Kapranov} [Duke Math. J. 71, No. 3, 633-664 (1993; Zbl 0804.14007), theorem 7.2] about logarithmic bundles.