Local enumerative invariants of some Gorenstein surfaces and orbifolds

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2023

Author dc:creator

• Nam, Sungwoo

Contributors dc:contributor

• Katz, Sheldon
• Duursma, Iwan
• Pascaleff, James
• Dodd, Christopher

• 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

Statement dc:rights

• Copyright 2023 Sungwoo Nam

Language dc:language

en, eng

Handle dc:identifier

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

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F121490