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On the equations defining some Hilbert schemes

Abstract : We work out details of the extrinsic geometry for two Hilbert schemes of some contemporary interest: the Hilbert scheme of two points on the projective plane and the dense open set parametrizing non-planar clusters in the punctual Hilbert scheme of clusters of length four on affine three-space with support at the origin. We find explicit equations in natural projective, respectively affine embeddings for these spaces. In particular, we answer a question of Bernd Sturmfels who asked for a description of the latter space that is amenable to further computations. While the explicit equations we find are controlled in a precise way by the representation theory of SL_3, our arguments also rely on computer algebra.
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Preprints, Working Papers, ...
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https://hal-cnrs.archives-ouvertes.fr/hal-03514369
Contributor : Laurent Manivel Connect in order to contact the contributor
Submitted on : Thursday, January 6, 2022 - 11:44:05 AM
Last modification on : Friday, January 7, 2022 - 3:57:11 AM

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  • HAL Id : hal-03514369, version 1
  • ARXIV : 2103.16363

Citation

Jonathan D. Hauenstein, Laurent Manivel, Balazs Szendroi. On the equations defining some Hilbert schemes. 2022. ⟨hal-03514369⟩

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