Combinatorial methods in algebraic geometry

Abstract

In this thesis, we study how to use combinatorics to solve problems in algebraic geometry. We succeed in our goal in many different areas. Firstly, we obtain results about normality and Gorenstein property of the varieties associated to phylogenetic group-based models. In particular, we prove a conjecture of Micha\l ek about the normality of the 3-Kimura model and extend results of Buczynska and Wisniewski about the Gorenstein property from the group Z_2 to groups Z_3 and Z_2xZ_2. We also obtain a full classification of graphical matroids whose associated varieties satisfy the Gorenstein property. Moreover, we classify tangential varieties to Segre-Veronese varieties which are Gorenstein or Cohen-Macaulay. Finally, we find formulas for computing the intersection products in the space of complete quadrics. We find the connection between these numbers, the maximum-likelihood degree of general linear concentration models, and the degree of semidefinite programming. These results allow us to prove Nie-Ranestadt-Sturmfels conjecture about the degree of semidefinite programming and also Sturmfels-Uhler conjecture about the polynomiality of maximum-likelihood degree.