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Birational Geometry of the Space of Complete Quadrics

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thesis
posted on 20.06.2014 by Cesar A. Lozano Huerta
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.

History

Advisor

Coskun, Izzet

Department

Mathematics, Statistics, and Computer Science

Degree Grantor

University of Illinois at Chicago

Degree Level

Doctoral

Committee Member

Ein, Lawrence Popa, Mihnea Huizenga, Jack De Fernex, Tommaso

Submitted date

2014-05

Language

en

Issue date

20/06/2014

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