Local enumerative invariants of some Gorenstein surfaces and orbifolds

Nam, Sungwoo

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https://hdl.handle.net/2142/121490 Copy

Description

Title

Local enumerative invariants of some Gorenstein surfaces and orbifolds

Author(s)

Nam, Sungwoo

Issue Date

2023-07-11

Director of Research (if dissertation) or Advisor (if thesis)

Katz, Sheldon

Doctoral Committee Chair(s)

Duursma, Iwan

Committee Member(s)

• Pascaleff, James
• Dodd, Christopher

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Local Gromov-Witten theory
• Local Gopakumar-Vafa theory
• simple normal crossing surfaces
• del Pezzo surfaces
• shrinkable surfaces
• root stacks
• moduli space of stable sheaves on root stacks

Abstract

In this thesis, we study questions from enumerative geometry of local surfaces. We give a definition for local Gromov–Witten and local Gopakumar–Vafa invariants for some singular surfaces as a generalization of the local theory of smooth del Pezzo surfaces. The singular surfaces we consider are motivated by the physics of 5d SCFTs. We also discuss the solution to the embeddability question of these surfaces. Unlike smooth surfaces, it is a subtle question to ask whether a surface S can be embedded into a smooth Calabi–Yau 3-fold. As a technical tool, we review the theory of root stacks and toric diagrams which give a Calabi–Yau 3-fold. The construction using root stacks leads to the local theory of root stacks. The motivation for this direction is provided from the crepant resolution conjecture. We also study the Hilbert scheme of 0-dimensional substacks on a root stack of a surface along a smooth divisor and compute its topological Euler characteristic.

Graduation Semester

2023-08

Type of Resource

Thesis

Copyright and License Information

Copyright 2023 Sungwoo Nam

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