On the equations defining some Hilbert schemes

Jonathan D. Hauenstein; Laurent Manivel; Balazs Szendroi

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arXiv · 2103.16363

On the equations defining some Hilbert schemes

Jonathan D. Hauenstein·Laurent Manivel·Balazs Szendroi

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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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Jonathan D. Hauenstein, Laurent Manivel, Balazs Szendroi. 2021-03-30. On the equations defining some Hilbert schemes. https://arxiv.org/abs/2103.16363

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