On the equations defining some Hilbert schemes
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...
| Main Authors: | , , |
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| Format: | Journal article |
| Language: | English |
| Published: |
Springer
2022
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| Summary: | 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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