Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scharlau, Rudolf / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

4 Issues per year

IMPACT FACTOR 2017: 0.734

CiteScore 2017: 0.70

SCImago Journal Rank (SJR) 2017: 0.695
Source Normalized Impact per Paper (SNIP) 2017: 0.891

Mathematical Citation Quotient (MCQ) 2017: 0.62

See all formats and pricing
More options …
Volume 18, Issue 4


Commuting matrices and the Hilbert scheme of points on affine spaces

Abdelmoubine A. Henni
  • Universidade Federal de Santa Catarina, Departamento de Matemática, Campus Universitário Trindade, CEP 88.040-900 Florianápolis-SC, Brasil
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Marcos Jardim
  • Corresponding author
  • IMECC - UNICAMP, Departamento de Matemática, Rua Sérgio Buarque de Holanda, 651, Cidade Universitária, 13083-859 Campinas–SP, Brazil
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2018-07-20 | DOI: https://doi.org/10.1515/advgeom-2018-0011


We give linear algebraic and monadic descriptions of the Hilbert scheme of points on the affine space of dimension n which naturally extends Nakajima’s representation of the Hilbert scheme of points on the plane. As an application of our ideas and recent results from the literature on commuting matrices, we show that the Hilbert scheme of c points on ℂ3 is irreducible for c ≤ 10.

Keywords: Hilbert scheme of points; monads

MSC 2010: 14C05


  • [1]

    A. Álvarez, F. Sancho, P. Sancho, Homogeneous Hilbert scheme. Proc. Amer. Math. Soc. 136 (2008), 781–790. MR2361849 Zbl 1131.14008Google Scholar

  • [2]

    D. A. Cartwright, D. Erman, M. Velasco, B. Viray, Hilbert schemes of 8 points. Algebra Number Theory 3 (2009), 763–795. MR2579394 Zbl 1187.14005Google Scholar

  • [3]

    M. Cirafici, A. Sinkovics, R. J. Szabo, Cohomological gauge theory, quiver matrix models and Donaldson-Thomas theory. Nuclear Phys. B 809 (2009), 452–518. MR2478118 Zbl 1192.81309Google Scholar

  • [4]

    G. Fløystad, Monads on projective spaces. Comm. Algebra 28 (2000), 5503–5516. MR1808585 Zbl 0977.14007Google Scholar

  • [5]

    J. Fogarty, Algebraic families on an algebraic surface. Amer. J. Math 90 (1968), 511–521. MR0237496 Zbl 0176.18401Google Scholar

  • [6]

    F. Galluzzi, F. Vaccarino, Hilbert-Chow morphism for non-commutative Hilbert schemes and moduli spaces of linear representations. Algebr. Represent. Theory 13 (2010), 491–509. MR2660858 Zbl 1203.14004Google Scholar

  • [7]

    M. Gerstenhaber, On dominance and varieties of commuting matrices. Ann. of Math. (2) 73 (1961), 324–348. MR0132079 Zbl 0168.28201Google Scholar

  • [8]

    A. Grothendieck, Techniques de construction et théorèmes ďexistence en géométrie algébrique. IV. Les schémas de Hilbert. In: Séminaire Bourbaki, Vol. 6, Exp. No. 221, 249–276, Soc. Math. France, Paris 1961. MR1611822 Zbl 0236.14003Google Scholar

  • [9]

    T. S. Gustavsen, D. Laksov, R. M. Skjelnes, An elementary, explicit, proof of the existence of Hilbert schemes of points. J. Pure Appl. Algebra 210 (2007), 705–720. MR2324602 Zbl 1122.14004Google Scholar

  • [10]

    R. Hartshorne, Connectedness of the Hilbert scheme. Inst. Hautes Études Sci. Publ. Math. no. 29 (1966), 5–48. MR0213368 Zbl 0171.41502Google Scholar

  • [11]

    A. A. Henni, M. Jardim, R. V. Martins, ADHM construction of perverse instanton sheaves. Glasg. Math. J. 57 (2015), 285–321. MR3333943 Zbl 1316.14024Google Scholar

  • [12]

    J. Holbrook, M. Z. Omladič, Approximating commuting operators. Linear Algebra Appl. 327 (2001), 131–149. MR1823346 Zbl 0978.15011Google Scholar

  • [13]

    A. Iarrobino, Reducibility of the family of 0-dimensional schemes on a variety. Inventiones Math. 15 (1972), 72–77. MR0301010 Zbl 0227.14006Google Scholar

  • [14]

    A. Iarrobino, Jr., Compressed algebras and components of the punctual Hilbert scheme. In: Algebraic geometry, Sitges (Barcelona), 1983, volume 1124 of Lecture Notes in Math., 146–165, Springer 1985. MR805334 Zbl 0567.14001Google Scholar

  • [15]

    M. Jardim, R. V. Martins, The ADHM variety and perverse coherent sheaves. J. Geom. Phys. 61 (2011), 2219–2232. MR2827120 Zbl 1229.14010Google Scholar

  • [16]

    A. D. King, Moduli of representations of finite-dimensional algebras. Quart. J. Math. Oxford Ser. (2) 45 (1994), 515–530. MR1315461 Zbl 0837.16005Google Scholar

  • [17]

    T. S. Motzkin, O. Taussky, Pairs of matrices with property L. II. Trans. Amer. Math. Soc. 80 (1955), 387–401. MR0086781 Zbl 0067.25401Google Scholar

  • [18]

    D. Mumford, Geometric invariant theory. Springer 1965. MR0214602 Zbl 0147.39304Google Scholar

  • [19]

    H. Nakajima, Lectures on Hilbert schemes of points on surfaces, volume 18 of University Lecture Series. Amer. Math. Soc. 1999. MR1711344 Zbl 0949.14001Google Scholar

  • [20]

    N. Nitsure, Construction of Hilbert and Quot schemes. In: Fundamental algebraic geometry, volume 123 of Math. Surveys Monogr., 105–137, Amer. Math. Soc. 2005. MR2223407 Zbl 1085.14001Google Scholar

  • [21]

    C. Okonek, M. Schneider, H. Spindler, Vector bundles on complex projective spaces. Birkhäuser 1980. MR561910 Zbl 0438.32016Google Scholar

  • [22]

    K. C. O’Meara, J. Clark, C. I. Vinsonhaler, Advanced topics in linear algebra. Oxford Univ. Press 2011. MR2849857 Zbl 1235.15013Google Scholar

  • [23]

    G. F. Seelinger, Brauer–Severi schemes of finitely generated algebras. Israel J. Math. 111 (1999), 321–337. MR1710744 Zbl 0964.16026Google Scholar

  • [24]

    C. S. Seshadri, Vector bundles on curves. In: Linear algebraic groups and their representations (Los Angeles, CA, 1992), volume 153 of Contemp. Math., 163–200, Amer. Math. Soc. 1993. MR1247504 Zbl 0799.14013Google Scholar

  • [25]

    K. Šivic, On varieties of commuting triples III. Linear Algebra Appl. 437 (2012), 393–460. MR2921710 Zbl 1323.15011Google Scholar

  • [26]

    F. Vaccarino, Linear representations, symmetric products and the commuting scheme. J. Algebra 317 (2007), 634–641. MR2362934 Zbl 1155.13007Google Scholar

  • [27]

    M. Van den Bergh, The Brauer-Severi scheme of the trace ring of generic matrices. In: Perspectives in ring theory (Antwerp, 1987), volume 233 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 333–338, Kluwer 1988. MR1048420 Zbl 0761.13003Google Scholar

About the article

Received: 2015-09-24

Revised: 2016-08-13

Published Online: 2018-07-20

Published in Print: 2018-10-25

Funding: AAH was supported by the FAPESP post-doctoral grant number 2009/12576-9. MJ is partially supported by the CNPq grant number 303332/2014-0 and the FAPESP grant number 2014/14743-8.

Citation Information: Advances in Geometry, Volume 18, Issue 4, Pages 467–482, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2018-0011.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in