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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 28th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexHave added a bunch of references to this entry. **Question:** What precisely can one say about the relation between the topological space underlying the Hilbert scheme of points of $\mathbb{C}$ and/or $\mathbb{C}^2$, and the [[Fulton-MacPherson compactification]] of the corresponding [[configuration spaces of points]]? There is commentary in just this direction on p. 189 of: * [[William Fulton]], [[Robert MacPherson]], _A compactification of configuration spaces_, Ann. of Math. (2), 139(1):183–225, 1994 ([jstor:2946631](https://www.jstor.org/stable/2946631)) but it remains unclear to me what exactly the statement is, in the end. <a href="https://ncatlab.org/nlab/revision/diff/Hilbert+scheme/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/Hilbert+scheme/6">v6</a>, <a href="https://ncatlab.org/nlab/show/Hilbert+scheme">current</a>

   Have added a bunch of references to this entry.

   Question: What precisely can one say about the relation between the topological space underlying the Hilbert scheme of points of $\mathbb{C}$ and/or $\mathbb{C}^2$, and the Fulton-MacPherson compactification of the corresponding configuration spaces of points?

   There is commentary in just this direction on p. 189 of:

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

   but it remains unclear to me what exactly the statement is, in the end.

   diff, v6, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 6th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to * Artan Sheshmani, _Hilbert Schemes, Donaldson-Thomas Theory, Vafa-Witten and Seiberg Witten theories_ ([arxiv:1911.01796](https://arxiv.org/abs/1911.01796)) <a href="https://ncatlab.org/nlab/revision/diff/Hilbert+scheme/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/Hilbert+scheme/7">v7</a>, <a href="https://ncatlab.org/nlab/show/Hilbert+scheme">current</a>

   added pointer to

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

   diff, v7, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeDec 29th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to discussion of the [[Hilbert scheme of points]] on [[K3-surfaces]] with an eye towards [[Rozansky-Witten theory]]: * [[Justin Sawon]], Section 5.3 of: _Rozansky-Witten invariants of hyperkähler manifold_, Cambridge 2000 ([arXiv:math/0404360](https://arxiv.org/abs/math/0404360)) <a href="https://ncatlab.org/nlab/revision/diff/Hilbert+scheme/8">diff</a>, <a href="https://ncatlab.org/nlab/revision/Hilbert+scheme/8">v8</a>, <a href="https://ncatlab.org/nlab/show/Hilbert+scheme">current</a>

   added pointer to discussion of the Hilbert scheme of points on K3-surfaces with an eye towards Rozansky-Witten theory:

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

   diff, v8, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeDec 29th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded these pointers: * [[Hiraku Nakajima]], _Lectures on Hilbert schemes of points on surfaces_, University Lecture Series, vol. 18, American Mathematical Society, Providence, RI, 1999 ([ams:ulect-18](https://bookstore.ams.org/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](https://arxiv.org/abs/1401.6782)) <a href="https://ncatlab.org/nlab/revision/diff/Hilbert+scheme/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/Hilbert+scheme/9">v9</a>, <a href="https://ncatlab.org/nlab/show/Hilbert+scheme">current</a>

   added these pointers:

   • 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)

   diff, v9, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeDec 29th 2019
   • (edited Dec 29th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointers to 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](https://arxiv.org/abs/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](https://arxiv.org/abs/hep-th/0309270)) * D. Maulik, A. Oblomkov, _Quantum cohomology of the Hilbert scheme of points on A_n-resolutions_, J. Amer. Math. Soc. 22 (2009), 1055-1091 ([arXiv:0802.2737](https://arxiv.org/abs/0802.2737)) See also: * Yukari Ito, [[Hiraku Nakajima]], _McKay correspondence and Hilbert schemes in dimension three_ ([math/9803120](https://arxiv.org/abs/math/9803120)) <a href="https://ncatlab.org/nlab/revision/diff/Hilbert+scheme/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/Hilbert+scheme/12">v12</a>, <a href="https://ncatlab.org/nlab/show/Hilbert+scheme">current</a>

   added pointers to 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 A_n-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)

   diff, v12, current

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeDec 30th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded references on 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](https://arxiv.org/abs/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](https://arxiv.org/abs/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](https://arxiv.org/abs/1309.2657)) * [[Alexander Braverman]], [[Michael Finkelberg]], [[Hiraku Nakajima]], _Line bundles over Coulomb branches_ ([arXiv:1805.11826](https://arxiv.org/abs/1805.11826)) * [[Ben Webster]], _Coherent sheaves on Hilbert schemes through the Coulomb lens_, 2018 ([[WebsterHilbertScheme18.pdf:file]]) * 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](https://arxiv.org/abs/1812.08788)) <a href="https://ncatlab.org/nlab/revision/diff/Hilbert+scheme/14">diff</a>, <a href="https://ncatlab.org/nlab/revision/Hilbert+scheme/14">v14</a>, <a href="https://ncatlab.org/nlab/show/Hilbert+scheme">current</a>

   added references on 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 (WebsterHilbertScheme18.pdf:file)

   • 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)

   diff, v14, current

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJan 2nd 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer on discussion in the context of [[Witten indices]] and [[K-theory|K-theoretic]] [[enumerative geometry]]: * [[Andrei Okounkov]], _Lectures on K-theoretic computations in enumerative geometry_ ([arXiv:1512.07363](https://arxiv.org/abs/1512.07363)) <a href="https://ncatlab.org/nlab/revision/diff/Hilbert+scheme/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/Hilbert+scheme/15">v15</a>, <a href="https://ncatlab.org/nlab/show/Hilbert+scheme">current</a>

   added pointer on 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)

   diff, v15, current

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJan 14th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to today's * [[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](https://arxiv.org/abs/2001.03834)) <a href="https://ncatlab.org/nlab/revision/diff/Hilbert+scheme/16">diff</a>, <a href="https://ncatlab.org/nlab/revision/Hilbert+scheme/16">v16</a>, <a href="https://ncatlab.org/nlab/show/Hilbert+scheme">current</a>

   added pointer to today’s

   • 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)

   diff, v16, current

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeJul 28th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[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](https://doi.org/10.1007/978-3-662-03276-3)) <a href="https://ncatlab.org/nlab/revision/diff/Hilbert+scheme/19">diff</a>, <a href="https://ncatlab.org/nlab/revision/Hilbert+scheme/19">v19</a>, <a href="https://ncatlab.org/nlab/show/Hilbert+scheme">current</a>

   added pointer to:

   • 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)

   diff, v19, current