Open Access
2016 On the functoriality of marked families
Paolo Lella, Margherita Roggero
J. Commut. Algebra 8(3): 367-410 (2016). DOI: 10.1216/JCA-2016-8-3-367


The application of methods of computational algebra has recently introduced new tools for the study of Hilbert schemes. The key idea is to define flat families of ideals endowed with a scheme structure whose defining equations can be determined by algorithmic procedures. For this reason, several authors developed new methods, based on the combinatorial properties of Borel-fixed ideals, that allow associating to each ideal $J$ of this type a scheme $\MFScheme {J}$, called a $J$-marked scheme. In this paper, we provide a solid functorial foundation to marked schemes and show that the algorithmic procedures introduced in previous papers do not depend on the ring of coefficients. We prove that, for all strongly stable ideals $J$, the marked schemes $\MFScheme {J}$ can be embedded in a Hilbert scheme as locally closed subschemes, and that they are open under suitable conditions on $J$. Finally, we generalize Lederer's result about Gr\"obner strata of zero-dimensional ideals, proving that Gr\"obner strata of any ideals are locally closed subschemes of Hilbert schemes.


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Paolo Lella. Margherita Roggero. "On the functoriality of marked families." J. Commut. Algebra 8 (3) 367 - 410, 2016.


Published: 2016
First available in Project Euclid: 9 September 2016

zbMATH: 1348.14010
MathSciNet: MR3546003
Digital Object Identifier: 10.1216/JCA-2016-8-3-367

Primary: 13P99 , 14C05

Keywords: Borel-fixed ideal , Hilbert scheme , marked family , open subfunctor

Rights: Copyright © 2016 Rocky Mountain Mathematics Consortium

Vol.8 • No. 3 • 2016
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