10027/18779 Cesar A. Lozano Huerta Cesar A. Lozano Huerta Birational Geometry of the Space of Complete Quadrics University of Illinois at Chicago 2014 algebraic gemeotry birational geometry complete quadrics minimal model program Mori's program Hassett-Keel program moduli spaces 2014-06-20 00:00:00 Thesis https://indigo.uic.edu/articles/thesis/Birational_Geometry_of_the_Space_of_Complete_Quadrics/10791467 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.