Abstract
Consider a finite scheme of length l contained in a smooth quadric surface over the complex numbers. We determine the number of linearly independent curves passing through the scheme, of degree at least \(l - 2\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Białynicki-Birula, A.: Some theorems on actions of algebraic groups. Ann. Math. 98, 480–497 (1973)
Carlini, E., Catalisano, M.V., Oneto, A.: On the Hilbert function of general fat points in \({\mathbb{P} }^1 \times {\mathbb{P} }^1\). Michigan Math. J. 69, 601–632 (2020)
Catalisano, M.V., Geramita, A.V., Gimigliano, A.: Higher secant varieties of Segre-Veronese varieties. In: Ciliberto, C., Geramita, A.V., et al. (eds.) Projective Varieties with Unexpected Properties. Walter de Gruyter, Berlin and New York (2005)
Durfee, A.H.: Algebraic varieties which are a disjoint union of subvarieties. In: McCrory, C., Shifrin, T. (eds.) Geometry and Topology. Manifolds, Varieties, and Knots. Marcel Dekker, New York (1987)
Ellingsrud, G., Strømme, S.A.: On the homology of the Hilbert scheme of points in the plane. Invent. Math. 87, 343–352 (1987)
Favacchio, G., Guardo, E.: On the Betti numbers of three fat points in \({\mathbb{P} }^1 \times {\mathbb{P} }^1\). J. Korean Math. Soc. 56, 751–766 (2019)
Fulton, W.: Intersection Theory, 2nd edn. Springer Verlag, Cham (1998)
Giuffrida, S., Maggioni, R., Ragusa, A.: On the postulation of \(0\)-dimensional subschemes on a smooth quadric. Pacific J. Math. 155, 251–282 (1992)
Giuffrida, S., Maggioni, R., Zappalà, G.: Scheme-theoretic complete intersections in \({\mathbb{P} }^1 \times {\mathbb{P} }^1\). Commun. Algebra 41, 532–551 (2013)
Göttsche, L.: The Betti numbers of the Hilbert scheme of points on a smooth projective surface. Math. Ann. 286, 193–207 (1990)
Guardo, E.: Fat points schemes on a smooth quadric. J. Pure Appl. Algebra 162, 183–208 (2001)
Guardo, E., Van Tuyl, A.: The minimal resolutions of double points in \({\mathbb{P} }^1 \times {\mathbb{P} }^1\) with ACM support. J. Pure Appl. Algebra 211, 784–800 (2007)
Guardo, E., Van Tuyl, A.: On the Hilbert functions of sets of points in \({\mathbb{P} }^1 \times {\mathbb{P} }^1 \times {\mathbb{P} }^1\). Math. Proc. Cambr. Philos. Soc. 159, 115–123 (2015)
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, 223–247 (2002)
Van Tuyl, A.: The Hilbert functions of ACM sets of points in \({\mathbb{P} }^{n_1} \times \dots \times {\mathbb{P} }^{n_k}\). J. Algebra 264, 420–441 (2003)
Acknowledgements
The author would like to thank the referee for suggesting several improvements to the presentation and for pointing out the bibliography concerning Hilbert functions of finite schemes in multiprojective spaces.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Maican, M. On the Hilbert function of a finite scheme contained in a quadric surface. Collect. Math. (2023). https://doi.org/10.1007/s13348-023-00406-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13348-023-00406-8