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.;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.;In Chapter 3, we study the following question: when does a model of X, defined as X(D) := Proj(⊕m≥0 H0(X,mD)), have a moduli interpretation? We describe such an interpretation for the models X(Hk) (Theorem B), where Hk is any generator of the nef cone 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 Mov(X), and among which we find a moduli interpretation for seven of them.;Chapter 4 outlines the relation of this work with that of Semple as well as future directions of research.
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