Birational Geometry of the Space of Complete Quadrics

Let $X$ be the moduli space of complete $(n-1)$-quadrics. In this thesis, we study the birational geometry of $X$ using tools from the minimal model program (MMP). In Chapter $1$, we recall the definition of the space $X$ and summarize our main results in Theorems A, B and C. \medskip In Chapter $2$, we examine the codimension-one cycles of the space $X$, and exhibit generators for Eff$(X)$ and Nef$(X)$ (Theorem A), the cone of effective divisors and the cone of nef divisors, respectively. This result, in particular, allows us to conclude the space $X$ is a Mori dream space. \medskip In Chapter $3$, we study the following question: when does a model of $X$, defined as $X(D):= \mathrm{Proj}(\bigoplus_{m\ge 0}H^0(X,mD))$, have a moduli interpretation? We describe such an interpretation for the models $X(H_k)$ (Theorem B), where $H_k$ is any generator of the nef cone $\mathrm{Nef}(X)$. In the case of complete quadric surfaces there are 11 birational models $X(D)$ (Theorem B), where $D$ is a divisor in the movable cone $\mathrm{Mov}(X)$, and among which we find a moduli interpretation for seven of them. \medskip Chapter 4 outlines the relation of this work with that of Semple \cite{SEM}, \cite{SEMII} as well as future directions of research.