The geometry of Hilbert schemes of two points on projective space

Ryan, Tim

doi:10.48550/arxiv.2103.12674

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2021

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“…which arises from the known intersection numbers [16] on the Hilbert scheme, namely (H 4 , H 3 B, H 2 B 2 , HB 3 , B 4 ) = (3, 0, −8, −24, −48).…”

Section: Introductionmentioning

confidence: 99%

On the equations defining some Hilbert schemes

Hauenstein¹,

Manivel²,

Szendrői³

2021

Preprint

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We work out details of the extrinsic geometry for two Hilbert schemes of some contemporary interest: the Hilbert scheme Hilb 2 P 2 of two points on P 2 and the dense open set parametrizing nonplanar clusters in the punctual Hilbert scheme Hilb 4 0 (A 3 ) of clusters of length four on A 3 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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“…which arises from the known intersection numbers [16] on the Hilbert scheme, namely (H 4 , H 3 B, H 2 B 2 , HB 3 , B 4 ) = (3, 0, −8, −24, −48).…”

Section: Introductionmentioning

confidence: 99%