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.

## References

### General

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

### Hilbert schemes of points

Specifically on Hilbert schemes of points:

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

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

#### As moduli spaces of instantons

Discussion in their role as moduli spaces of instantons:

• Sean Pohorence, Hilbert scheme of points and the ADHM construction (pdf)

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)