Abstract
Let \({Q = {\mathbb P}^1 \times {\mathbb P}^1}\) and let \({X\subset Q}\) be a zero-dimensional scheme. The results in this paper give the possibility of computing, under certain hypotheses, the Hilbert function of a zero-dimensional scheme in Q that is not ACM. In particular we show how, under some conditions on X, its Hilbert function changes when we add points to X lying on a (1, 0) or (0, 1)-line. As a particular case we show also that if X is ACM this result holds without any additional hypothesis.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Giuffrida S., Maggioni R.: Curves on a smooth quadric. Collect. Math. 54(3), 309–325 (2003)
Giuffrida S., Maggioni R., Ragusa A.: On the postulation of 0-dimensional subschemes on a smooth quadric. Pacific J. Math. 155(2), 251–282 (1992)
Guardo E.: Fat points schemes on a smooth quadric. J. Pure Appl. Algebra 162, 183–208 (2001)
Guardo E., Van Tuyl A.: Fat points in \({\mathbb P^1\times \mathbb P^1}\) and their Hilbert functions. Can. J. Math. 56(4), 716–741 (2004)
Van Tuyl A.: The border of the Hilbert function of a set of points in \({{\mathbb P}^{n_1}\times\dots\times{\mathbb P}^{n_k}}\) . J. Pure Appl. Algebra 176(2–3), 223–247 (2002)
Van Tuyl A.: The Hilbert function of ACM set of points in \({{\mathbb P}^{n_1}\times\dots\times{\mathbb P}^{n_k}}\) . J. Algebra 264, 420–441 (2003)
Van Tuyl, A.: An appendix to a paper of Catalisano, Geramita, Gimigliano: the Hilbert function of generic sets of 2-fat points in \({{\mathbb P}^1\times {\mathbb P}^1}\) . In: Projective Varieties with Unexpected Properties, pp. 109–112. de Gruyter, Berlin (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bonacini, P., Marino, L. On the Hilbert function of zero-dimensional schemes in \({{\mathbb P}^1 \times {\mathbb P}^1}\) . Collect. Math. 62, 57–67 (2011). https://doi.org/10.1007/s13348-010-0004-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13348-010-0004-x