The geometry of some moduli schemes arising in enumerative geometry
<p>In this thesis, we work on the explicit geometric description of certain moduli schemes. Pandharipande-Thomas (PT) stable pairs provide an example of moduli spaces of objects in the derived category, whose scheme-theoretic geometry can be explored to a reasonable level of details. We consid...
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| Médium: | Diplomová práce |
| Jazyk: | English |
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2022
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| _version_ | 1826312859003387904 |
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| author | Carlucci, A |
| author2 | Szendrői, B |
| author_facet | Szendrői, B Carlucci, A |
| author_sort | Carlucci, A |
| collection | OXFORD |
| description | <p>In this thesis, we work on the explicit geometric description of certain moduli schemes. Pandharipande-Thomas (PT) stable pairs provide an example of moduli spaces of objects in the derived category, whose scheme-theoretic geometry can be explored to a reasonable level of details. We consider PT-pairs supported at the double of the zero-section inside a particular bundle over the projective line, called the resolved conifold.</p>
<p>Their geometry can be probed in two ways. The first is sheaf-theoretic, and consists in looking at the non-reduced structures with which we can endow a reduced curve: this relies on a procedure by Ferrand, reducing the study to line bundles. Degenerations of those line bundles are relevant, as they carry information about the corresponding PT-pairs.</p>
<p>The second way involves representation theory. Thanks to a result by Nagao-Nakajima, PT-pairs on the resolved conifold correspond to stable representations of a particular quiver with potential, for a suitable stability condition. By finding an appropriate basis, we can read off the equations for the moduli scheme and recognise the same geometry observed through the first approach. </p>
<p>There is a stratification of the moduli scheme, which admits a simple sheaf-theoretic interpretation, but the actual equations are given via the quiver representa-tion approach. The main technical obstacle is that, while stability for the Hilbert scheme (leading to Donaldson-Thomas invariants) translates to a simple algebraic condition, stability of PT-pairs involves a case-by-case analysis to obtain suitable algebraic conditions on the corresponding quiver representations.</p> |
| first_indexed | 2024-09-25T04:03:32Z |
| format | Thesis |
| id | oxford-uuid:6768ae8d-f5cb-4fd6-a6fb-f391a7608d36 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-09-25T04:03:32Z |
| publishDate | 2022 |
| record_format | dspace |
| spelling | oxford-uuid:6768ae8d-f5cb-4fd6-a6fb-f391a7608d362024-05-09T09:51:57ZThe geometry of some moduli schemes arising in enumerative geometryThesishttp://purl.org/coar/resource_type/c_db06uuid:6768ae8d-f5cb-4fd6-a6fb-f391a7608d36Geometry, AlgebraicModuli theoryGeometry, EnumerativeEnglishHyrax Deposit2022Carlucci, ASzendrői, BKirwan, FThomas, R<p>In this thesis, we work on the explicit geometric description of certain moduli schemes. Pandharipande-Thomas (PT) stable pairs provide an example of moduli spaces of objects in the derived category, whose scheme-theoretic geometry can be explored to a reasonable level of details. We consider PT-pairs supported at the double of the zero-section inside a particular bundle over the projective line, called the resolved conifold.</p> <p>Their geometry can be probed in two ways. The first is sheaf-theoretic, and consists in looking at the non-reduced structures with which we can endow a reduced curve: this relies on a procedure by Ferrand, reducing the study to line bundles. Degenerations of those line bundles are relevant, as they carry information about the corresponding PT-pairs.</p> <p>The second way involves representation theory. Thanks to a result by Nagao-Nakajima, PT-pairs on the resolved conifold correspond to stable representations of a particular quiver with potential, for a suitable stability condition. By finding an appropriate basis, we can read off the equations for the moduli scheme and recognise the same geometry observed through the first approach. </p> <p>There is a stratification of the moduli scheme, which admits a simple sheaf-theoretic interpretation, but the actual equations are given via the quiver representa-tion approach. The main technical obstacle is that, while stability for the Hilbert scheme (leading to Donaldson-Thomas invariants) translates to a simple algebraic condition, stability of PT-pairs involves a case-by-case analysis to obtain suitable algebraic conditions on the corresponding quiver representations.</p> |
| spellingShingle | Geometry, Algebraic Moduli theory Geometry, Enumerative Carlucci, A The geometry of some moduli schemes arising in enumerative geometry |
| title | The geometry of some moduli schemes arising in enumerative geometry |
| title_full | The geometry of some moduli schemes arising in enumerative geometry |
| title_fullStr | The geometry of some moduli schemes arising in enumerative geometry |
| title_full_unstemmed | The geometry of some moduli schemes arising in enumerative geometry |
| title_short | The geometry of some moduli schemes arising in enumerative geometry |
| title_sort | geometry of some moduli schemes arising in enumerative geometry |
| topic | Geometry, Algebraic Moduli theory Geometry, Enumerative |
| work_keys_str_mv | AT carluccia thegeometryofsomemodulischemesarisinginenumerativegeometry AT carluccia geometryofsomemodulischemesarisinginenumerativegeometry |