Smooth Hilbert schemes: Their classification and geometry
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Abstract
Closed subschemes in projective space with a fixed Hilbert polynomial are parametrized by a Hilbert scheme. We classify the smooth ones. We identify numerical conditions on a polynomial that completely determine when the Hilbert scheme is smooth. We also reinterpret these smooth Hilbert schemes as generalized partial flag varieties and describe the subschemes being parametrized.
Funding statement: Gregory G. Smith was partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Knut and Alice Wallenberg Foundation.
Acknowledgements
We thank Dave Anderson, Sam Payne, Mike Roth, Mike Stillman, and an anonymous referee for their suggestions. Computational experiments done in Macaulay2 [10] were indispensable.
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Articles in the same Issue
- Frontmatter
- Arc-descent for the perfect loop functor and p-adic Deligne–Lusztig spaces
- A Landau–Ginzburg mirror theorem via matrix factorizations
- Singularity models of pinched solutions of mean curvature flow in higher codimension
- Borel subgroups of the plane Cremona group
- Monoidal abelian envelopes with a quotient property
- Weakly non-collapsed RCD spaces are strongly non-collapsed
- Lower bounds for the scalar curvatures of Ricci flow singularity models
- Smooth complex projective rational surfaces with infinitely many real forms
- Smooth Hilbert schemes: Their classification and geometry
Articles in the same Issue
- Frontmatter
- Arc-descent for the perfect loop functor and p-adic Deligne–Lusztig spaces
- A Landau–Ginzburg mirror theorem via matrix factorizations
- Singularity models of pinched solutions of mean curvature flow in higher codimension
- Borel subgroups of the plane Cremona group
- Monoidal abelian envelopes with a quotient property
- Weakly non-collapsed RCD spaces are strongly non-collapsed
- Lower bounds for the scalar curvatures of Ricci flow singularity models
- Smooth complex projective rational surfaces with infinitely many real forms
- Smooth Hilbert schemes: Their classification and geometry
