Geometry of logarithmic mapping spaces in genus one

In this thesis, we explore several parts of the geometry of the moduli spaces of genus one logarithmic stable maps. In Chapter 2, we exhibit a smooth compactification of the moduli space of elliptic curves in a product of projective spaces with tangency along a subset of its toric boundary divisors. This is a Vakil–Zinger type of desingularization for maps to a product of projective spaces using ideas of elliptic singularities and logarithmic geometry. However, we also give an explicit example showing that, when the target is P¹ × P¹, the Vakil–Zinger operations alone are not enough to desingularize the space; we do so by comparing the absolute geometry with the logarithmic one. Using the smoothness result, we construct the virtual fundamental classes of moduli spaces of maps to a special class of simple normal crossings pairs. We then prove a consistency result that relates the virtual classes when we remove certain “fictitious” markings, suggesting the theory has good recursive properties. In Chapter 3, we study the desingularized moduli spaces of genus 1 maps to projective spaces in closer detail. This allows us to describe the class of a stratum in the logarithmic moduli spaces by performing tautological operations on the corresponding stratum in the moduli spaces of genus 1 stable maps. We show there is an analogue of the splitting formula in the context of genus 1 logarithmic Gromov–Witten theory, which lays the groundwork for logarithmic degeneration formulae in this reduced genus one logarithmic Gromov–Witten theory.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2022

Author dc:creator

• Zheng, Wanlong

Advisor dc:contributor.advisor

• Ranganathan, Dhruv

• algebraic geometry
• moduli spaces

Licence

• https://www.rioxx.net/licenses/all-rights-reserved/

Language dc:language

eng

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.93102

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/345680