Contando curvas: uma aplicação da geometria tropical

Abstract

The main objective of this work is to present some techniques from tropical geometry for counting the number of algebraic curves in the complex projective plane of genus g and degree d, which pass through 3d + g − 1 points in general position, denoted by Ncplx(d, g). To do this, we will first introduce some notions of algebraic geometry necessary for understanding the work. Then we will present the Caporaso-Harris formula, used to calculate Ncplx(d, g). Next, we will introduce tropical curves, defining the tropical semibody and tropical hypersurfaces, and prove the Duality Theorem, which relates tropical curves to subdivisions of their Newton polygon. We will also show tropical curves as balanced graphs. Afterwards, we will define the λ-crescent lattice paths and explain their relationship with the calculation of Ncplx(d, g), using tropical curves. Finally, we will show that reticulated paths also satisfy the Caporaso-Harris Formula, which is the main result of our work.