nLab Hilbert scheme

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 K3-surface

2. 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.

Related concepts

• homotopy of rational maps

References

General

Textbook account:

• János Kollár, Chapter I of: Rational Curves on Algebraic Varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete 32, Springer 1996 (doi:10.1007/978-3-662-03276-3)

See also

• Wikipedia, Hilbert scheme

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:

• William Fulton, Robert MacPherson, p. 189 of: A compactification of configuration spaces, Ann. of Math. (2), 139(1):183–225, 1994 (jstor:2946631)

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:

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

• Ian Grojnowski, Instantons and affine algebras I: The Hilbert scheme and vertex operators (arXiv:alg-geom/9506020)

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)

See also

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

Of ADE singularities

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

• Kenji Mohri, Kähler Moduli Space of a D-Brane at Orbifold Singularities, Commun. Math. Phys. 202 (1999) 669-699 (arXiv:hep-th/9806052)

• Ron Donagi, Sheldon Katz, Eric Sharpe, Spectra of D-branes with Higgs vevs, Adv.Theor.Math.Phys. 8 (2005) 813-859 (arXiv:hep-th/0309270)

• D. Maulik, A. Oblomkov, Quantum cohomology of the Hilbert scheme of points on An-resolutions_, J. Amer. Math. Soc. 22 (2009), 1055-1091 (arXiv:0802.2737)

See also:

• Yukari Ito, Hiraku Nakajima, McKay correspondence and Hilbert schemes in dimension three (math/9803120)

Of K3 surfaces

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

• Arnaud Beauville, Variétés Kähleriennes dont la premiere classe de Chern est nulle, Jour.

  Diff. Geom. 18 (1983), 755–782 (euclid.jdg/1214438181)

• Georg Oberdieck, Gromov-Witten invariants of the Hilbert schemes of points of a K3 surface, Geom. Topol. 22 (2018) 323-437 (arXiv:1406.1139, thesis pdf)

  (using Gromov-Witten invariants)

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:

• Cumrun Vafa, Edward Witten, Section 4.1 of: A Strong Coupling Test of S-Duality, Nucl. Phys. B431:3-77, 1994 (arXiv:hep-th/9408074)

• Cumrun Vafa, Instantons on D-branes, Nucl. Phys. B463 (1996) 435-442 (arXiv:hep-th/9512078)

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

• Jeffrey Harvey, Gregory Moore, Section 5.1 of: On the algebras of BPS states, Commun. Math. Phys. 197:489-519, 1998 (arXiv:hep-th/9609017)

• Kazuyuki Furuuchi, Hiroshi Kunitomo, Toshio Nakatsu, p. 17 of: Topological Field Theory and Second-Quantized Five-Branes, Nucl.Phys. B494 (1997) 144-160 (arXiv:hep-th/9610016)

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):

• Justin Roberts, Simon Willerton, p. 17 of: On the Rozansky-Witten weight systems, Algebr. Geom. Topol. 10 (2010) 1455-1519 (arXiv:math/0602653)

• Justin Sawon, Section 5.3 of: Rozansky-Witten invariants of hyperkähler manifold, Cambridge 2000 (arXiv:math/0404360)

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:

• Jan de Boer, Kentaro Hori, Hirosi Ooguri, Yaron Oz, Mirror Symmetry in Three-Dimensional Gauge Theories, Quivers and D-branes, Nucl. Phys. B493:101-147, 1997 (arXiv:hep-th/9611063)

• Jan de Boer, Kentaro Hori, Hirosi Ooguri, Yaron Oz, Zheng Yin, Mirror Symmetry in Three-Dimensional Gauge Theories, $SL(2,\mathbb{Z})$ and D-Brane Moduli Spaces, Nucl. Phys. B493:148-176, 1997 (arXiv:hep-th/9612131)

• Stefano Cremonesi, Amihay Hanany, Alberto Zaffaroni, sround (4.4) of: Monopole operators and Hilbert series of Coulomb branches of 3d $\mathcal{N} = 4$ gauge theories, JHEP 01 (2014) 005 (arXiv:1309.2657)

• Alexander Braverman, Michael Finkelberg, Hiraku Nakajima, Line bundles over Coulomb branches (arXiv:1805.11826)

• Ben Webster, Coherent sheaves on Hilbert schemes through the Coulomb lens, 2018 (pdf)

• Mykola Dedushenko, Yale Fan, Silviu Pufu, Ran Yacoby, Section E.2 of: Coulomb Branch Quantization and Abelianized Monopole Bubbling, JHEP 10 (2019) 179 (arXiv:1812.08788)

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

• Andrei Okounkov, Lectures on K-theoretic computations in enumerative geometry (arXiv:1512.07363)