Quiver varieties and moduli spaces of sheaves on singular surfaces
<p>We investigate several types of Nakajima quiver varieties and their connection to the Kleinian singularities 𝐂<sup>2</sup>/Γ.</p> <p>Such quiver varieties have many good properties: they are for instance irreducible and have symplectic singularities, in particular...
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| Other Authors: | |
| Format: | Thesis |
| Language: | English |
| Published: |
2022
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| _version_ | 1797108462425997312 |
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| author | Gammelgaard, S |
| author2 | Szendrői, B |
| author_facet | Szendrői, B Gammelgaard, S |
| author_sort | Gammelgaard, S |
| collection | OXFORD |
| description | <p>We investigate several types of Nakajima quiver varieties and their connection to the Kleinian singularities 𝐂<sup>2</sup>/Γ.</p>
<p>Such quiver varieties have many good properties: they are for instance irreducible and have symplectic singularities, in particular they are normal.</p>
<p>The first part of this thesis introduces Nakajima quiver varieties, together with the necessary background on projective Deligne-Mumford stacks and framed sheaves. Following that, we prove that the punctual Hilbert schemes (when taken with their reduced scheme structures) Hilb<sup>n</sup>(𝐂<sup>2</sup>/Γ) are examples of Nakajima quiver varieties.</p>
<p>We then show that there is a type of 'orbifold Quot scheme' generalising the Hilbert scheme, which can also be identified with a Nakajima quiver variety.</p>
<p>Following this, we investigate another generalisation of the Hilbert scheme: that of a moduli space of framed sheaves on a certain stack compactifying 𝐂<sup>2</sup>/Γ. We show that this moduli space exists as a quasiprojective scheme. We are unable to show that it is isomorphic to a Nakajima quiver variety, but we show that it carries a canonical morphism to a Nakajima quiver variety, and this morphism is a bijection of closed points.</p>
<p>We end by sketching some potential further directions of investigation.</p> |
| first_indexed | 2024-03-07T07:28:04Z |
| format | Thesis |
| id | oxford-uuid:e45a0d8b-76cb-4089-9e28-90dd0e5cb759 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:28:04Z |
| publishDate | 2022 |
| record_format | dspace |
| spelling | oxford-uuid:e45a0d8b-76cb-4089-9e28-90dd0e5cb7592022-12-06T17:00:09ZQuiver varieties and moduli spaces of sheaves on singular surfacesThesishttp://purl.org/coar/resource_type/c_db06uuid:e45a0d8b-76cb-4089-9e28-90dd0e5cb759Geometry, AlgebraicEnglishHyrax Deposit2022Gammelgaard, SSzendrői, B <p>We investigate several types of Nakajima quiver varieties and their connection to the Kleinian singularities 𝐂<sup>2</sup>/Γ.</p> <p>Such quiver varieties have many good properties: they are for instance irreducible and have symplectic singularities, in particular they are normal.</p> <p>The first part of this thesis introduces Nakajima quiver varieties, together with the necessary background on projective Deligne-Mumford stacks and framed sheaves. Following that, we prove that the punctual Hilbert schemes (when taken with their reduced scheme structures) Hilb<sup>n</sup>(𝐂<sup>2</sup>/Γ) are examples of Nakajima quiver varieties.</p> <p>We then show that there is a type of 'orbifold Quot scheme' generalising the Hilbert scheme, which can also be identified with a Nakajima quiver variety.</p> <p>Following this, we investigate another generalisation of the Hilbert scheme: that of a moduli space of framed sheaves on a certain stack compactifying 𝐂<sup>2</sup>/Γ. We show that this moduli space exists as a quasiprojective scheme. We are unable to show that it is isomorphic to a Nakajima quiver variety, but we show that it carries a canonical morphism to a Nakajima quiver variety, and this morphism is a bijection of closed points.</p> <p>We end by sketching some potential further directions of investigation.</p> |
| spellingShingle | Geometry, Algebraic Gammelgaard, S Quiver varieties and moduli spaces of sheaves on singular surfaces |
| title | Quiver varieties and moduli spaces of sheaves on singular surfaces |
| title_full | Quiver varieties and moduli spaces of sheaves on singular surfaces |
| title_fullStr | Quiver varieties and moduli spaces of sheaves on singular surfaces |
| title_full_unstemmed | Quiver varieties and moduli spaces of sheaves on singular surfaces |
| title_short | Quiver varieties and moduli spaces of sheaves on singular surfaces |
| title_sort | quiver varieties and moduli spaces of sheaves on singular surfaces |
| topic | Geometry, Algebraic |
| work_keys_str_mv | AT gammelgaards quivervarietiesandmodulispacesofsheavesonsingularsurfaces |