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Logarithmic sheaves attached to arrangements of hyperplanes, preprint
"... Any divisor D on a nonsingular variety X defines a sheaf of logarithmic differential forms Ω1 X (log D). Its equivalent definitions and many useful properties are discussed in a fundamental paper of K. Saito [Sa]. This sheaf is locally free when D is a strictly normal crossing divisor, and in this s ..."
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Any divisor D on a nonsingular variety X defines a sheaf of logarithmic differential forms Ω1 X (log D). Its equivalent definitions and many useful properties are discussed in a fundamental paper of K. Saito [Sa]. This sheaf is locally free when D is a strictly normal crossing divisor, and in this situation
On a compactification of the moduli space of the rational normal curves
"... Abstract. For any odd n, we construct a smooth minimal (i.e. obtained by adding an irreducible hypersurface) compactification Mn of the quasiprojective homogeneous variety Sn = P GL(n + 1)/ SL(2) that parameterizes the rational normal curves in P n . Mn is isomorphic to a component of the Maruyama ..."
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Abstract. For any odd n, we construct a smooth minimal (i.e. obtained by adding an irreducible hypersurface) compactification Mn of the quasiprojective homogeneous variety Sn = P GL(n + 1)/ SL(2) that parameterizes the rational normal curves in P n . Mn is isomorphic to a component of the Maruyama scheme of the semistable sheaves on P n of rank n and Chern polynomial (1 + t) n+2 . This will allow us to explicitly compute the Betti numbers of Mn. In particular M 3 is isomorphic to the variety of nets of quadrics defining twisted cubics, studied by G. Ellinsgrud, R. Piene and S. Strømme [EPS].
Adv. Geom. 1 (2001), 165±192 Advances in Geometry ( de Gruyter 2001 Unstable hyperplanes for Steiner bundles and multidimensional matrices
"... Ottaviani* (Communicated by G. Gentili) Abstract. We study some properties of the natural action of SL
V0 SL
Vp on nondegenerate multidimensional complex matrices A A P
V0 n nVp of boundary format (in the sense of Gelfand, Kapranov and Zelevinsky); in particular we characterize th ..."
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Ottaviani* (Communicated by G. Gentili) Abstract. We study some properties of the natural action of SL
V0 SL
Vp on nondegenerate multidimensional complex matrices A A P
V0 n nVp of boundary format (in the sense of Gelfand, Kapranov and Zelevinsky); in particular we characterize the nonstable ones as the matrices which are in the orbit of a ``triangular' ' matrix, and the matrices with a stabilizer containing C as those which are in the orbit of a ``diagonal' ' matrix. For p 2 it turns out that a nondegenerate matrix A A P
V0 nV1 nV2 detects a Steiner bundle SA (in the sense of Dolgachev and Kapranov) on the projective space Pn, n dim
V2 ÿ 1. As a consequence we prove that the symmetry group of a Steiner bundle is contained in SL
2 and that the SL
2invariant Steiner bundles are exactly the bundles introduced by Schwarzenberger [Schw], which correspond to ``identity' ' matrices. We can characterize the points of the moduli space of Steiner bundles which are stable for the action of Aut
Pn, answering in the ®rst nontrivial case a question posed by Simpson. In the opposite direction we obtain some results about Steiner bundles which imply properties of matrices. For example the number of unstable hyperplanes of SA (counting multiplicities) produces an interesting discrete invariant of A, which can take the values 0; 1; 2;...; dim V0 1 or y; the y case occurs if and only if SA is Schwarzenberger (and A is an identity). Finally, the Gale transform for Steiner bundles introduced by Dolgachev and Kapranov under the classical name of association can be understood in this setting as the transposition operator on multidimensional matrices. 1