Abstract
Hilbert schemes of zerodimensional ideals in a polynomial ring can be covered with suitable affine open subschemeswhose construction is achieved using border bases. Moreover, border bases have proved to be an excellent tool for describing zero-dimensional ideals when the coefficients are inexact. and in this situation they show a clear advantage with respect to Gröbner bases which, nevertheless, can also be used in the study of Hilbert schemes, since they provide tools for constructing suitable stratifications.
In this paper we compare Gröbner basis schemes with border basis schemes. It is shown that Gröbner basis schemes and their associated universal families can be viewed as weighted projective schemes. A first consequence of our approach is the proof thatall the ideals which define a Gröbner basis scheme and are obtained using Buchberger's Algorithm, are equal. Another result is that if the origin (i.e. the point corresponding to the unique monomial ideal) in the Gröbner basis scheme is smooth, then the scheme itself is isomorphic to an affine space. This fact represents a remarkable difference between border basis and Gröbner basis schemes. Since it is natural to look for situations where a Gröbner basis scheme and the corresponding border basis scheme are equal, we address the issue, provide an answer, and exhibit some consequences. Open problems are discussed at the end of the paper.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D. Bayer,The division algorithm and the Hilbert scheme, Thesis, Harvard University Cambridge, MA, 1982.
The CoCoA Team, CoCoA:a system for doing Computations in Commutative Algebra, available at http://cocoa.dima.unige.it.
A. Conca and nG. Valla, Canonical Hilbert-Burch matrices for ideals ofk[x, y], arXiv:math/0708.3576.
D. Eisenbud,Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics 150, SpringerVerlag, New York, 1995.
T.S. Gustavsen, D. Laksov, and R.M. Skjelnes, An elementary, explicit proof of the existence of Hilbert schemes of points,J. Pure Appl. Algebra 210 (2007), 705–720.
M. Haiman,q,t-Catalan numbers and the Hilbert scheme,Discre1te Math. 193 (1998), 201–224.
M.E. Huibregtse, A description of certain affine open subschemes that form an open covering of\(Hilb_{\mathbb{A}_k^2 }^n \),Pacific J. Math. 204 (2002), 97–143.
M.E. Huibregtse, An elementary construction of the multigraded Hilbert scheme of points,Pacific J. Math. 223 (2006), 269–315.
M.E. Huibregtse, The cotangent space at a monomial ideal of the Hilbert scheme of points of an affine space, arXiv:math/0506575.
A. Kehrein and M. Kreuzer, Characterizations of border bases,J. Pure Appl. Alg. 196 (2005), 251–270.
M. Kreuzer and L. Robbiano,Computational Commutative Algebra, 1, SpringerVerlag, Berlin, 2000.
M. Kreuzer and L. Robbiano,Computational Commutative Algebra, 2, SpringerVerlag, Berlin, 2005.
M. Kreuzer and L. Robbiano, Deformations of border bases,Collect. Math. 59 (2008), 275–297.
E. Miller and B. Sturmfels,Combinatorial Commutative Algebra, Graduate Texts in Mathematics 277, SpringerVerlag, New York, 2005.
B. Mourrain,A New Criterion for Normal Form Algorithms, Lecture Notes in Comput. Sci. 1719 SpringerVerlag, Berlin, 1999.
R. Notari and M.L. Spreafico, A stratification of Hilbert schemes by initial ideals and applications,Manuscripta Math. 101 (2000), 429–448.
S. Onn and B. Sturmfels, Cutting corners,Adv. in Appl. Math. 23 (1999), 29–48.
H.J. Stetter,Numerical Polynomial Algebra, Society for Industrial and Applied Mathematics, Philadephia, PA, 2004.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Robbiano, L. On border basis and Gröbner basis schemes. Collect. Math. 60, 11–25 (2009). https://doi.org/10.1007/BF03191213
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF03191213