Resultants of determinantal varieties
type: Item
instance of: scholarly article
title: Resultants of determinantal varieties
zbMATH Open document ID: 1057.14064
author: Laurent Busé
published in: Journal of Pure and Applied Algebra
publication date: 2004-08-19T00:00:00Z
review text: 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)].
reviewed by: Isidro Nieto Baños
Mathematics Subject Classification ID: 14M12
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 14Q20
zbMATH DE Number: 2092906
zbMATH Keywords: effectivity
zbMATH Keywords: determinantal resultant
zbMATH Keywords: mixed resultant
zbMATH Keywords: Macaulay resultant
zbMATH Keywords: syzygies and resolutions
MaRDI profile type: Publication
full work available at URL: j.jpaa.2004.02.010
OpenAlex ID: W2029625397
cites work: Q3347989
cites work: Q3789692
cites work: The Geometry of Schemes
cites work: Resultants and Chow forms via exterior syzygies
cites work: Q4293510
cites work: Q4143433
cites work: Le formalisme du résultant. (The formalism of resultant)
cites work: Inertia forms and resultant: A formulary
cites work: Chow polytopes and general resultants
cites work: Syzygies des variétés déterminantales
cites work: Faisceaux algébriques cohérents
cites work: Q4850733
DOI: 10.1016/J.JPAA.2004.02.010
Recommended article: Explicit formulas for the multivariate resultant.
Recommended article: Q4318627
Recommended article: Q4722194
Recommended article: On a result of Gelfand, Kapranov and Zelevinsky
Recommended article: Q3135195
Recommended article: Resultant over the residual of a complete intersection.
Recommended article: Determinantal hypersurfaces.
Recommended article: Chern classes and characteristic cycles of determinantal varieties
Recommended article: Matrix formulæ for resultants and discriminants of bivariate tensor-product polynomials
Recommended article: A package for computations with classical resultants
description: scientific article; zbMATH DE number 2092906