Enumerative aspects of the Gross-Siebert program
Michel van Garrel, D. Peter Overholser, Helge Ruddat
TL;DR
This paper surveys the enumerative aspects of the Gross-Siebert program, highlighting the role of tropical geometry in counting algebraic curves and exploring mirror symmetry applications, especially for the projective plane.
Contribution
It provides an introductory overview of the Gross-Siebert program, including definitions, results, and applications of tropical geometry to enumerative problems and mirror symmetry.
Findings
Counting algebraic curves via tropical curves
Examples illustrating tropical geometry methods
Application to mirror symmetry of the projective plane
Abstract
We present enumerative aspects of the Gross-Siebert program in this introductory survey. After sketching the program's main themes and goals, we review the basic definitions and results of logarithmic and tropical geometry. We give examples and a proof for counting algebraic curves via tropical curves. To illustrate an application of tropical geometry and the Gross-Siebert program to mirror symmetry, we discuss the mirror symmetry of the projective plane.
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Taxonomy
TopicsAdvanced Differential Equations and Dynamical Systems · Polynomial and algebraic computation · Algebraic Geometry and Number Theory