Resultants of determinantal varieties
In this paper based on part of the author's doctoral dissertation, the author aims to extend the so called mixed resultant of \textit{I. M. Gelfand, M. M. Kapranov} and \textit{A. V. Zelevinsky} [Discriminants, resultants and multidimensional determinants. Birkhäuser, Berlin (1994; Zbl 0827.14036)] of complete intersection varieties to the context of determinantal varieties.~In section 2, he defines the \(r\)th-determinantal resultant.~Being given two vector bundles E and F of respective ranks \(m \geq n \) on an irreducible projective variety \(X\)~ and an integer \( 0 \leq r <n \) the author gives in theorem 1 suitable hypothesis on the dimension of \(X\)~ and on the vector bundles \(E\) and \(F\) such that the determinantal resultant exists.~Then in theorem 3, it is shown that as for the resultant of a very ample vector bundle as in loc.cit, one can compute the degree (and the multi-degree if E splits) of the determinant resultant in terms of the Chern classes of \(E\) and \(F\). In section 3, the author recalls the resolution of a determinantal variety following \textit{A. Lascoux} [Adv. Math. 30, 202--237 (1978; Zbl 0394.14022)]. It is used in section 4 to show that in theorem 14, always as for the resultant of Gelfand et al in loc. cit, the determinantal resultant can be computed as the determinant of a certain complex.~In section 5, an algorithm is given to compute explicitely the determinantal resultant, completely similar to the one giving the expression of the Macaulay resultant (see prop. 16).~In section 5.3, the classical Sylvester resultants and Dixon resultants are generalized to the determinantal case.~The author concludes with prop.~17 computing the Chow form of a rational normal scroll.~Such formulas are already well known as e.g. by \textit{D. Eisenbud} and \textit{F.-O. Schreyer} [J. Am. Math. Soc. 16, 537--579 (2003; Zbl 1069.14019)].
scientific article; zbMATH DE number 2092906
scientific article