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.
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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.
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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.
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Chapter 4 outlines the relation of this work with that of Semple \cite{SEM}, \cite{SEMII} as well as future directions of research.