Full Description
This book is based on the lectures given at the Oberwolfach Seminar held in Fall 2021. Logarithmic Gromov-Witten theory lies at the heart of modern approaches to mirror symmetry, but also opens up a number of new directions in enumerative geometry of a more classical flavour. Tropical geometry forms the calculus through which calculations in this subject are carried out. These notes cover the foundational aspects of this tropical calculus, geometric aspects of the degeneration formula for Gromov-Witten invariants, and the practical nuances of working with and enumerating tropical curves. Readers will get an assisted entry route to the subject, focusing on examples and explicit calculations.
Contents
Part I: Toric Geometry and Logarithmic Curve Counting. - 1. Geometry of Toric Varieties. - 2. Compactifying Subvarieties of Tori. - 3. Points on the Riemann Sphere. - 4. Stable Maps and Logarithmic Stable Maps. - 5. Cheat Codes for Logarithmic GW Theory. - Part II: Hurwitz Theory. - 6. Classical Hurwitz Theory and Moduli Spaces. - 7. Tropical Hurwitz Theory. - 8. Hurwitz Numbers from Piecewise Polynomials. - Part III: Tropical Plane Curve Counting. - 9. Introduction to Plane Tropical Curve Counts. - 10. Lattice Paths and the Caporaso-Harris Formula. - 11. The Caporaso-Harris Formula for Tropical Plane Curves and Floor Diagrams.
