Algebraic Methods in Reconstruction of Varieties and Game Theory

Abstract

In this thesis we study the synergies among theoretical, computational and applied algebraic geometry in three different settings, that correspond to each of the chapters: plane Hurwitz numbers, Hessian correspondence and algebraic game theory.

First, we study plane Hurwitz numbers from a theoretical and computational point of view. We explore how to reconstruct $\mathfrak{h}_d$ plane curves from the branch locus of the projection from a fix point, where $\mathfrak{h}_d$ is the plane Hurwitz number of degree $d$. We approach this recovery problem for the case of plane cubics and plane quartics. From a theoretical point of view, we compute the real plane Hurwitz numbers $\hr_3$ and $\hr_4$. Moreover, we introduce and study the notion of (real) Segre-Hurwitz numbers $\mathfrak{sh}_{d_1,d_2}$ and $\mathfrak{sh}_{d_1,d_2}^\mathrm{real}$.

Secondly, we investigate the Hessian correspondence of hypersurfaces $H_{d,n}$. This is the map that associate to a degree $d$ hypersurface in $\P^n$ its Hessian variety or second polar variety. We analyse the fibers of $H_{d,n}$ and its birationality for hypersurfaces of Waring rank at most $n+1$ and for hypersurfaces of degree $3$ and $4$. From a computational perspective, we explore how to recover a hypersurface from its Hessian variety. We also investigate the geometry of the catalecticant enveloping variety.

Thirdly, we explore the application of algebro-geometric tools to the study of the conditional independence (CI) equilibria of a collection of CI statements $\mathcal{C}$. We focus on the case where $\mathcal{C}$ is the global Markov property of an undirected graph. We investigate this notion of equilibrium through the algebro-geometric examination of the Spohn CI variety. We restrict our study to binary games and we analyse the dimension of these varieties. In the case where the graph is a disjoint union of complete graphs, we explore further algebro-geometric features of Spohn CI varieties.