Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Families of determinantal schemes

Authors: Jan O. Kleppe and Rosa M. Miró-Roig
Journal: Proc. Amer. Math. Soc. 139 (2011), 3831-3843
MSC (2010): Primary 14M12, 14C05, 14H10, 14J10
Published electronically: March 16, 2011
MathSciNet review: 2823030
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Given integers $ a_0\le a_1\le \cdots \le a_{t+c-2}$ and $ b_1\le \cdots \le b_t$, we denote by $ W(\underline{b};\underline{a})\subset \textrm{Hilb}^p(\mathbb{P}^{n})$ the locus of good determinantal schemes $ X\subset \mathbb{P}^{n}$ of codimension $ c$ defined by the maximal minors of a $ t\times (t+c-1)$ homogeneous matrix with entries homogeneous polynomials of degree $ a_j-b_i$. The goal of this paper is to extend and complete the results given by the authors in an earlier paper and determine under weakened numerical assumptions the dimension of $ W(\underline{b};\underline{a})$ as well as whether the closure of $ W(\underline{b};\underline{a})$ is a generically smooth irreducible component of $ \textrm{Hilb}^p(\mathbb{P}^{n})$.

References [Enhancements On Off] (What's this?)

  • 1. A. Beauville, Determinantal hypersurfaces, Michigan Math. J. 48, no. 1 (2000), 39-64. MR 1786479 (2002b:14060)
  • 2. W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, 1993. MR 1251956 (95h:13020)
  • 3. W. Bruns and U. Vetter, Determinantal rings, Lecture Notes in Mathematics, 1327, Springer-Verlag, New York/Berlin, 1988. MR 953963 (89i:13001)
  • 4. D.A. Buchsbaum and D. Eisenbud, What annihilates a module? J. Algebra 47 (1977), 231-243. MR 0476736 (57:16293)
  • 5. M.C. Chang, A filtered Bertini-type theorem, Crelle J. 397 (1989), 214-219. MR 993224 (90i:14054)
  • 6. D. Eisenbud, Commutative Algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995. MR 1322960 (97a:13001)
  • 7. G. Ellingsrud, Sur le schéma de Hilbert des variétés de codimension $ 2$ dans $ \mathbb{P}^{e}$ a cône de Cohen-Macaulay, Ann. Scient. Éc. Norm. Sup. 8 (1975), 423-432. MR 0393020 (52:13831)
  • 8. D. Grayson and M. Stillman.
    Macaulay 2--a software system for algebraic geometry and commutative algebra, available at .
  • 9. A. Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux, North-Holland, Amsterdam, 1968. MR 0476737 (57:16294)
  • 10. J.O. Kleppe, Families of low dimensional determinantal schemes. J. Pure Appl. Alg., online 9.11.2010, DOI: 101016/j.jpaa.2010.10.007.
  • 11. J.O. Kleppe, J. Migliore, R.M. Miró-Roig, U. Nagel and C. Peterson, Gorenstein liaison, complete intersection liaison invariants and unobstructedness, Memoirs Amer. Math. Soc. 154, no. 732, 2001.MR 1848976 (2002e:14083)
  • 12. J.O. Kleppe and R.M. Miró-Roig, Dimension of families of determinantal schemes, Trans. Amer. Math. Soc. 357 (2005), 2871-2907. MR 2139931 (2006b:14086)
  • 13. J. Kreuzer, J. Migliore, U. Nagel and C. Peterson, Determinantal schemes and Buchsbaum-Rim sheaves, J. Pure Appl. Algebra 150 (2000), 155-174. MR 1765869 (2001f:14092)
  • 14. M. Martin-Deschamps and R. Piene, Arithmetically Cohen-Macaulay curves in $ \mathbb{P}^4$ of degree $ 4$ and genus 0, Manuscripta Math. 93 (1997), 391-408. MR 1457737 (99b:14001)
  • 15. R.M. Miró-Roig, Determinantal ideals. Progress in Mathematics, 264, Birkhäuser Verlag, Basel, 2008. MR 2375719 (2008k:14098)
  • 16. A. Siqveland, Generalized matric Massey products for graded modules. J. Gen. Lie Theory Appl., in press.
  • 17. B. Ulrich, Ring of invariants and linkage of determinantal ideals, Math. Ann. 274 (1986), 1-17. MR 834101 (87d:14043)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 14M12, 14C05, 14H10, 14J10

Retrieve articles in all journals with MSC (2010): 14M12, 14C05, 14H10, 14J10

Additional Information

Jan O. Kleppe
Affiliation: Faculty of Engineering, Oslo University College, Pb. 4 St. Olavs plass, N-0130 Oslo, Norway

Rosa M. Miró-Roig
Affiliation: Departament d’Algebra i Geometria, Facultat de Matemàtiques, Universitat Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain

Received by editor(s): November 10, 2009
Received by editor(s) in revised form: September 17, 2010
Published electronically: March 16, 2011
Additional Notes: The second author was partially supported by MTM2010-15256.
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2011 American Mathematical Society

American Mathematical Society