Singularities of hyperdeterminants

Jerzy Weyman; Andrei Zelevinsky

Annales de l'institut Fourier (1996)

  • Volume: 46, Issue: 3, page 591-644
  • ISSN: 0373-0956

Abstract

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We study the singular locus of the variety of degenerate hypermatrices of an arbitrary format. Our main result is a classification of irreducible components of the singular locus. Equivalently, we classify irreducible components of the singular locus for the projectively dual variety of a product of several projective spaces taken in the Segre embedding.

How to cite

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Weyman, Jerzy, and Zelevinsky, Andrei. "Singularities of hyperdeterminants." Annales de l'institut Fourier 46.3 (1996): 591-644. <http://eudml.org/doc/75190>.

@article{Weyman1996,
abstract = {We study the singular locus of the variety of degenerate hypermatrices of an arbitrary format. Our main result is a classification of irreducible components of the singular locus. Equivalently, we classify irreducible components of the singular locus for the projectively dual variety of a product of several projective spaces taken in the Segre embedding.},
author = {Weyman, Jerzy, Zelevinsky, Andrei},
journal = {Annales de l'institut Fourier},
keywords = {hyperdeterminant; singular locus; cusp type singularities; node type singularities; projectively dual variety; Segre embedding},
language = {eng},
number = {3},
pages = {591-644},
publisher = {Association des Annales de l'Institut Fourier},
title = {Singularities of hyperdeterminants},
url = {http://eudml.org/doc/75190},
volume = {46},
year = {1996},
}

TY - JOUR
AU - Weyman, Jerzy
AU - Zelevinsky, Andrei
TI - Singularities of hyperdeterminants
JO - Annales de l'institut Fourier
PY - 1996
PB - Association des Annales de l'Institut Fourier
VL - 46
IS - 3
SP - 591
EP - 644
AB - We study the singular locus of the variety of degenerate hypermatrices of an arbitrary format. Our main result is a classification of irreducible components of the singular locus. Equivalently, we classify irreducible components of the singular locus for the projectively dual variety of a product of several projective spaces taken in the Segre embedding.
LA - eng
KW - hyperdeterminant; singular locus; cusp type singularities; node type singularities; projectively dual variety; Segre embedding
UR - http://eudml.org/doc/75190
ER -

References

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  2. [2] A. DIMCA, Milnor numbers and multiplicities of dual varieties, Rev. Roumaine Math. Pures Appl., 31 (1986), 535-538. Zbl0606.14002MR87k:14002
  3. [3] W. FULTON, J. HARRIS, Representation Theory, Graduate Texts in Mathematics, N° 129, Springer-Verlag, 1991. Zbl0744.22001MR93a:20069
  4. [4] I.M. GELFAND, M.M. KAPRANOV, A.V. ZELEVINSKY, Hyperdeterminants, Adv. in Math., 96 (1992), 226-263. Zbl0774.15002MR94g:14023
  5. [5] I.M. GELFAND, M.M. KAPRANOV, A.V. ZELEVINSKY, Discriminants, Resultants and Multidimensional Determinants, Birkhäuser, Boston, 1994. Zbl0827.14036
  6. [6] M.M. KAPRANOV, B. STURMFELS, A.V. ZELEVINSKY, Chow polytopes and general resultants, Duke Math. J., 67, N° 1 (1992), 189-218. Zbl0780.14027MR93e:14062
  7. [7] N. KATZ, Pinceaux de Lefschetz; Théorème d'existence, in : SGA 7, Lecture Notes in Math., vol. 340, 212-253. Zbl0284.14006
  8. [8] S. KLEIMAN, Enumerative theory of singularities, in : Real and complex singularities (Proc. Ninth Nordic Summer School / NAVF Sympos. Math., Oslo, 1976), 297-396. Zbl0385.14018MR58 #27960
  9. [9] I. MACDONALD, Symmetric functions and Hall polynomials, Clarendon Press, Oxford, 1979. Zbl0487.20007MR84g:05003
  10. [10] A. PARUSIŃSKI, Multiplicity of the dual variety, Bull. London Math. Soc., 23 (1991), 429-436. Zbl0714.14027MR93a:14006
  11. [11] L. SCHLÄFLI, Über die Resultante eines Systems mehrerer algebraischer Gleichungen, Denkschr. der Kaiserl. Akad. Wiss., Math-Naturwiss. Klasse, 4 (1852), reprinted in : Gessamelte Abhandlungen, vol.2, N° 9, p. 9-112, Birkhäuser-Verlag, Basel, 1953. 
  12. [12] J. WEYMAN, Calculating discriminants by higher direct images, Trans. AMS, 343, N° 1 (1994), 367-389. Zbl0823.14040MR94g:14026
  13. [13] J. WEYMAN, A.V. ZELEVINSKY, Multiplicative properties of projectively dual varieties, Manuscripta Math., 82 (1994), 139-148. Zbl0839.14039MR94m:14070

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