Abstract
In this paper we prove the dimension and the irreduciblity of the variety parametrizing all pairs of commuting nilpotent matrices. Our proof uses the connection between this variety and the punctual Hilbert scheme of a smooth algebraic surface.
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[BC] P. Bala, R. W. Carter,Classes of unipotent elements in simple algebraic groups, I and II. Math. Proc. Cambridge Philos. Soc.79 (1976), No. 3, 401–425, and80 (1976), No. 1, 1–17.
[Ba1] V. Baranovsky,On punctual quot schemes for algebraic surfaces, to appear in J. Differential Geometry, also preprint alg-geom/9703038.
[Ba2] V. Baranovsky,Moduli of sheaves on surfaces and action of the oscillator algebra, preprint math.AG/9811092
[B] J. Briançon,Description de Hilbn ℂ{x,y}, Invent. Math.41 (1977), No 1, 45–89.
[EL] G. Ellingsrud, M. Lehn,On the irreducibility of the punctual quotient scheme of a surface, Ark. Mat.37 (1999), No. 2, 245–254.
[F1] J. Fogarty,Truncated hilbert functors, J. reine angew. Math.234 (1969), 65–88.
[F2] J. Fogarty,Algebraic families on an algebraic surface. II.The Picard scheme of the punctual Hilbert scheme, Amer. J. Math.95 (1973), 660–687.
[GL] T. Gaffney, R. Lazarsfeld,On the ramification of branched coverings of P n, Invent. Math.59 (1980), 53–58.
[GM] M. Goresky, R. MacPherson,Intersection homology. II, Invent. Math.72 (1983), No. 1, 77–129.
[Gu] R. Guralnick,A note on commuting pairs of matrices, Linear and Multilinear Algebra31 (1992), No. 1–4, 71–75.
[Gra] M. Granger,Géométrie des Schémas de Hilbert Ponctuels, Mém. Soc. Math. France (N.S.)8 (1983), 84 pp.
[Gr] D. Gross,Higher nullcones and commuting varieties, Comm. Algebra21 (1993), No. 5, 1427–1455.
[G] A. Grothendieck,Techniques de construction et théorémes d'existence en géométrie algébrique IV:Les schémas de Hilbert, Séminaire Bourbaki221 (1960/61).
[GS] L. Göttsche, W. Soergel,Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces, Math. Ann.296 (1993), No. 2, 235–245.
[I] A. Iarrobino,Punctual Hilbert Schemes, Mem. Am. Math. Soc.188 (1977), 112 pp.
[MT] T. S. Motzkin, O. Taussky,Pairs of matrices with property L. II, Trans. Amer. Math. Soc.80 (1955), 387–401.
[N] H. Nakajima,Lectures on Hilbert Schemes of Points on Surfaces, University Lecture Series18. Amer. Math. Soc., Providence, RI, 1999.
[P] В. С. Пясецкий, Линейные группы Ли, дейсмвующие с конечным числом орбим, Функц. анализ и его прилож.9 (1975), No4, 85–86. English translation: V. S. Pyasetskii,Linear Lie groups acting with finitely many orbits, Funct. Anal. Appl.9 (1975), 351–353.
[Ri] R. W. Richardson,Commuting varieties of semisimple Lie algebras and algebraic groups, Compositio Math.38 (1979), No. 3, 311–327.
[S] S. A. Strømme,Elementary Introduction to Representable Functors and Hilbert Schemes, inParameter Spaces, Banach Center Publications36 (1996).
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Baranovsky, V. The variety of pairs of commuting nilpotent matrices is irreducible. Transformation Groups 6, 3–8 (2001). https://doi.org/10.1007/BF01236059
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DOI: https://doi.org/10.1007/BF01236059