Contents

# Contents

## Idea

Hilbert schemes are moduli spaces of subvarieties, hence configuration spaces in algebraic geometry.

For instance a scheme of 0-dimensional sub-schemes is called a Hilbert scheme of points, etc.

Specifically for quasi-projective variety with fixed Hilbert polynomial?, Hilbert schemes are well behave as moduli spaces go, in that they’re actually quasi-projective varieties themselves.

The existence and construction of Hilbert schemes is due to Grothendieck (FGA).

The Hilbert scheme of $\mathbb{C}^2$ is widely studied in combinatorics and geometric representation theory for its connections to Macdonald polynomials and Cherednik algebras.

## Properties

### Compact hyperkähler structure

The only known examples of compact hyperkähler manifolds are Hilbert schemes of points $X^{[n+1]}$ (for $n \in \mathbb{N}$) for $X$ either

1. a 4-torus (in which case the compact hyperkähler manifolds is really the fiber of $(\mathbb{T}^4)^{[n]} \to \mathbb{T}^4$)

(Beauville 83) and two exceptional examples (O’Grady 99, O’Grady 03 ), see Sawon 04, Sec. 5.3.

## References

### Hilbert schemes of points

• Hiraku Nakajima, Lectures on Hilbert schemes of points on surfaces, University Lecture Series, vol. 18, American Mathematical Society, Providence, RI, 1999 (ams:ulect-18)

• Hiraku Nakajima, More lectures on Hilbert schemes of points on surfaces, Advanced Studies in Pure Mathematics 69, 2016, Development of Moduli Theory – Kyoto 2013, 173-205 (arXiv:1401.6782)

• J. Bertin, The punctual Hilbert scheme: An introduction (pdf)

• Dori Bejleri, Hilbert schemes: Geometry, combionatorics, and representation theory (pdf)

• Barbara Bolognese, Ivan Losev, A general introduction to the Hilbert scheme of points on the plane (pdf)

• Joachim Jelisiejew, Pathologies on the Hilbert scheme of points (arXiv:1812.08531)

Discussion in relation to the Fulton-MacPherson compactifications of configuration spaces of points:

Discussion of Euler numbers of Hilbert schemes of points:

• Hiraku Nakajima, Euler numbers of Hilbert schemes of points on simple surface singularities and quantum dimensions of standard modules of quantum affine algebras (arXiv:2001.03834)

### As moduli spaces of instantons

Discussion in their role as moduli spaces of instantons:

Specifically in relation to Donaldson-Thomas theory:

• Michele Cirafici, Annamaria Sinkovics, Richard Szabo, Cohomological gauge theory, quiver matrix models and Donaldson-Thomas theory, Nucl. Phys. B809: 452-518, 2009 (arXiv:0803.4188)

• Artan Sheshmani, Hilbert Schemes, Donaldson-Thomas theory, Vafa-Witten and Seiberg-Witten theories, Notices of the International Congress of Chines Mathematics (2019) (j.mp:2U7qd01, pdf)

• Artan Sheshmani, Hilbert Schemes, Donaldson-Thomas Theory, Vafa-Witten and Seiberg Witten theories (arxiv:1911.01796)

• Jian Zhou, K-Theory of Hilbert Schemes as a Formal Quantum Field Theory (arXiv:1803.06080)

Discussion of the Hilbert schemes of points of ADE-singularities:

### Of K3 surfaces

Discussion of the Hilbert schemes of points of K3-surfaces:

Discussion of configuration spaces of possibly coincident points on K3-surfaces $X$, hence of symmetric products $X^n/Sym(n)$ as moduli spaces of D0-D4-brane bound states wrapped on K3-surfaces:

Suggestion that this is to be resolved by the Hilbert scheme of points:

Hilbert schemes on K3 as moduli space of stable vector bundles:

• Laura Costa Farràs, K3 surfaces: moduli spaces and Hilbert schemes, Collectanea Mathematica, 1998, vol. 49, núm. 2-3, p. 273-282 (hdl:2445/16925

with an eye towards Rozansky-Witten theory (ground field-valued Rozansky-Witten weight systems):

### Hilbert schemes and Higgs/Coulomb branches

Identification of Higgs branches/Coulomb branches in D=3 N=4 super Yang-Mills theory with Hilbert schemes of points of complex curves:

Discussion in the context of Witten indices and K-theoretic enumerative geometry:

Last revised on February 2, 2021 at 05:47:56. See the history of this page for a list of all contributions to it.