Toric degenerations, Fano schemes and computations in tropical geometry

Tropical geometry is a developing area of mathematics in between algebraic geometry, combinatorics and polyhedral geometry. The main objects of study are tropical varieties which can be seen as polyhedral and combinatorial shadows of classical algebraic varieties. These simpler geometric objects keep important geometric information. They can be used to tackle problems in algebraic geometry but also to develop a purely tropical theory which has connection to different areas of research. \n \nIn Chapter 3 we study toric degenerations of the full flag varieties Fl4 and Fl5 using their tropicalization and we compare these with toric degenerations coming from representation theory techniques, namely degenerations associated to string polytopes and the Feigin-Fourier-Littelmann-Vinberg polytope. \n \nThe classical Fano scheme Fd(X) of a projective variety X ⊂ Pn parametrises d-dimensional linear spaces contained in X. In Chapter 4 we study tropical versions of the Fano scheme and their relations with the classical Fd(X). One version is the tropicalization trop Fd(X) while the second Fd(trop X) has a completely tropical construction. An interesting problem is to understand when these two versions coincide. We address this problem for some specific varieties such as linear spaces, toric varieties and hypersurfaces. \n \nComputations with tropical varieties are at the basis of tropical geometry. In Chapter 5 we present a Macaulay2 package Tropical.m2 that we developed in order to provide a user friendly tool to do these computations in Macaulay2. \n