Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-25T05:55:26.874Z Has data issue: false hasContentIssue false
  • English
  • Français

Hilbert series of residual intersections

Part of: Cycles and subschemes Theory of modules and ideals Commutative algebra: Homological methods Local rings and semilocal rings Projective and enumerative geometry Special varieties

Published online by Cambridge University Press:  09 June 2015

Marc Chardin ,
David Eisenbud  and
Bernd Ulrich
Marc Chardin
Affiliation:
Institut de Mathématiques de Jussieu, CNRS and UPMC, 4 place Jussieu, 75005 Paris, France email [email protected]
David Eisenbud
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720, USA email [email protected]
Bernd Ulrich
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We give explicit formulas for the Hilbert series of residual intersections of a scheme in terms of the Hilbert series of its conormal modules. In a previous paper, we proved that such formulas should exist. We give applications to the number of equations defining projective varieties and to the dimension of secant varieties of surfaces and three-folds.

Keywords

residual intersectionHilbert seriesmultiplicityHilbert coefficientsconormal modulessecant varieties

MSC classification

Primary: 13D40: Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 13C40: Linkage, complete intersections and determinantal ideals
Secondary: 13H15: Multiplicity theory and related topics 14M06: Linkage 14C17: Intersection theory, characteristic classes, intersection multiplicities 14N15: Classical problems, Schubert calculus
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 9 , September 2015 , pp. 1663 - 1687
Copyright
© The Authors 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Apéry, R., Sur les courbes de première espèce de l’espace à trois dimensions, C. R. Acad. Sci. Paris Sér. I 220 (1945), 271–272.Google Scholar
Artin, M. and Nagata, M., Residual intersections in Cohen–Macaulay rings, J. Math. Kyoto Univ. 12 (1972), 307–323.Google Scholar
Avramov, L. and Herzog, J., The Koszul algebra of a codimension 2 embedding, Math. Z. 175 (1980), 249–260.CrossRefGoogle Scholar
Buchweitz, R., Contributions à la théorie des singularités, Thèse d’Etat, Université Paris 7 (1981). Available at https://tspace.library.utoronto.ca/handle/1807/16684.Google Scholar
Cartan, H. and Eilenberg, S., Homological algebra (Princeton University Press, Princeton, NJ, 1956).Google Scholar
Chardin, M., Eisenbud, D. and Ulrich, B., Hilbert functions, residual intersections, and residually S 2 ideals, Compositio Math. 125 (2001), 193–219.CrossRefGoogle Scholar
Cumming, C., Residual intersections in Cohen–Macaulay rings, J. Algebra 308 (2007), 91–106.CrossRefGoogle Scholar
Dale, M., Severi’s theorem on the Veronese surface, J. Lond. Math. Soc. (2) 32 (1985), 419–425.CrossRefGoogle Scholar
Eisenbud, D. and Evans, E. G., Generating modules efficiently: theorems from algebraic K-theory, J. Algebra 27 (1973), 278–305.CrossRefGoogle Scholar
Gaeta, F., Quelques progrès récents dans la classification des variétés algébriques d’un espace projectif, in Deuxième Colloq. de Géometrie Algébrique, Liège (Masson & Cie, Paris, 1952), 145–183.Google Scholar
Herzog, J., Komplexe, Auflösungen und Dualität in der lokalen Algebra, Habilitations-schrift, Universität Regensburg (1974).Google Scholar
Herzog, J., Simis, A. and Vasconcelos, W. V., Koszul homology and blowing-up rings, in Commutative algebra, Lecture Notes in Pure and Applied Mathematics, vol. 84, eds Greco, S. and Valla, G. (Marcel Dekker, New York, 1983), 79–169.Google Scholar
Huneke, C., Linkage and Koszul homology of ideals, Amer. J. Math. 104 (1982), 1043–1062.CrossRefGoogle Scholar
Huneke, C., Strongly Cohen–Macaulay schemes and residual intersections, Trans. Amer. Math. Soc. 277 (1983), 739–763.CrossRefGoogle Scholar
Huneke, C. and Martin, H., Residual intersections and the number of equations defining projective varieties, Comm. Algebra 23 (1995), 2345–2376.CrossRefGoogle Scholar
Kunz, E. and Waldi, R., Regular differential forms, Contemp. Math. 79 (1988).Google Scholar
Simis, A. and Ulrich, B., On the ideal of an embedded join, J. Algebra 226 (2000), 1–14.CrossRefGoogle Scholar
Stückrad, J., On quasi-complete intersections, Arch. Math. 58 (1992), 529–538.CrossRefGoogle Scholar
Ulrich, B., Remarks on residual intersections, in Free resolutions in commutative algebra and algebraic geometry, Sundance, 1990, Research Notes in Mathematics, vol. 2, eds Eisenbud, D. and Huneke, C. (Jones and Bartlett, Boston–London, 1992), 133–138.Google Scholar
Ulrich, B., Artin–Nagata properties and reductions of ideals, Contemp. Math. 159 (1994), 373–400.CrossRefGoogle Scholar
Watanabe, J., A note on Gorenstein rings of embedding codimension three, Nagoya Math. J. 50 (1973), 227–232.CrossRefGoogle Scholar