Hilbert schemes of 8 points
By Dustin A. Cartwright, Daniel Erman, Mauricio Velasco, B. Viray. 2008
111 citations recorded by the source corpus.
The Hilbert scheme H^d_n of n points in A^d contains an irreducible component R^d_n which generically represents n distinct points in A^d. We show that when n is at most 8, the Hilbert scheme H^d_n is reducible if and only if n = 8 and d >= 4. In the simplest case of reducibility, the component R^4_8 \subset H^4_8 is defined by a single explicit equation which serves as a criterion for deciding whether a given ideal is a limit of distinct points. To understand the components of the Hilbert scheme, we study the closed subschemes of H_n^d which parametrize those ideals which are homogeneous and have a fixed Hilbert function. These subschemes are a special case of multigraded Hilbert schemes, and we describe their components when the colength is at most 8. In particular, we show that the scheme corresponding to the Hilbert function (1,3,2,1) is the minimal reducible example.
Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/4275b10d50bbbb667f1cd03c2c2f0ecd1cd60992
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Citations in the corpus
- Isomorphism classes of short Gorenstein local rings via Macaulay's inverse system (2012, unknown): “[4].”
- Irreducibility of the Gorenstein loci of Hilbert schemes via ray families (2014, methodology): “To see that the forms satisfying the assumptions of the proposition constitute an irreducible subset of Ps we observe that every 0 as above is smoothable by [Cartwright et al. 2009].”