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An elementary, explicit, proof of the existence of Hilbert schemes of points
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2007 (English)In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 210, no 3, 705-720 p.Article in journal (Refereed) Published
Abstract [en]

There are many beautiful constructions of the Hilbert scheme of closed subschemes of projective schemes. In many cases where Hilbert schemes are involved, it suffices to know they exist. On the other hand, there are many situations when it is crucial to have an explicit description of the Hilbert schemes. In this article we give a simple construction of Hilbert schemes of points, under general circumstances, that provides such a description. In addition to being useful for computations the construction is short, elementary, and explicit. Our methods rely on simple algebraic constructions. It gives explicit expressions of members of an affine covering of the Hilbert schemes, involving few variables satisfying natural equations. In particular, it explains the connections with commuting schemes of matrices. We also give some examples showing how our method can be used.

Place, publisher, year, edition, pages
2007. Vol. 210, no 3, 705-720 p.
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line
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URN: urn:nbn:se:kth:diva-16702DOI: 10.1016/j.jpaa.2006.11.003ISI: 000247292900009Scopus ID: 2-s2.0-34247874742OAI: oai:DiVA.org:kth-16702DiVA: diva2:334745
Note
QC 20100525Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2017-12-12Bibliographically approved

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