Abstract
In this paper we extend work of Gerstenhaber [5], Kostant [9], Kraft-Procesi [10], Tanisaki [15], and others on orbit closures in the nilpotent cone of matrices by studying varieties of square matrices defined by conditions on the ranks of powers of the matrices, or more generally on the ranks of polynomial functions of them. We show that the irreducible components of such varieties are always Gorenstein with rational singularities (in particular they are normal). We compute their tangent spaces, and also their limits under deformations of the defining polynomial functions. We also study generators for the ideals of such varieties, and we compute the singular loci of the hypersurfaces in the space of n × n matrices given by the vanishing of a single coefficient of the characteristic polynomial.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
D. Bayer and M. Stillman, Macaulay, a computer algebra program, Available free from the authors, for many machines including the Macintosh, IBM-PC, Sun, Vax, and others (1986).
W. Borho and H.-P. Kraft, Über Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen, Comment. Math. Helv. 54 (1979), 61–104.
C. De Concini and C. Procesi, Symmetrie functions, conjugacy classes, and the flag variety, Invent. Math. 64 (1981), 203–219.
R. Elkik, Singularités rationelles et déformations, Invent. Math. 47 (1978), 139–147.
M. Gerstenhaber, On dominance and varieties of commuting matrices, Ann. of Math. (2) 73 (1961), 324–348.
H. Grauert and O. Riemenschneider, Verschwindungssätze für analytische Koho- mologiegruppen auf komplexen Räumen, Invent. Math. 11 (1970), 263–292.
R. Hartshorne, “Algebraic Geometry,” Springer-Ver lag, New York, 1977.
G. Kempf, F. Knudson, D. Mumford and B. Saint-Donat, “Toroidal embeddings, I,” Springer Lecture Notes in Math., vol. 338, 1973.
B. Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404.
H.-P. Kraft and C. Procesi, Closures of conjugacy classes of matrices are normal, Invent. Math. 53 (1979), 227–247.
T. Matsumura, T., “Commutative Algebra,” Benjamin/Cummings Publishing Co., 1980.
C. Procesi, C., A formal inverse to the Cay ley-Hamilton Theorem, preprint (1986).
I. R. Shafarevich, “Basic Algebraic Geometry,” Springer-Verlag, Berlin/Heidelberg/ New York, 1977.
E. Strickland, On the variety of projectors, J. Alg. 106 (1987), 135–147.
T. Tanisaki, Defining ideals of the closures of the conjugacy classes and representations of the Weyl groups, Tohoku Math. J. 34 (1982), 575–585.
J. Weyman, Equations of conjugacy classes of nilpotent matrices, preprint (1987).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1989 Springer-Verlag New York Inc.
About this paper
Cite this paper
Eisenbud, D., Saltman, D. (1989). Rank Varieties of Matrices. In: Hochster, M., Huneke, C., Sally, J.D. (eds) Commutative Algebra. Mathematical Sciences Research Institute Publications, vol 15. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-3660-3_9
Download citation
DOI: https://doi.org/10.1007/978-1-4612-3660-3_9
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-8196-2
Online ISBN: 978-1-4612-3660-3
eBook Packages: Springer Book Archive