- No file added yet -

# Birational Geometry of the Space of Complete Quadrics

thesis

posted on 2014-06-20, 00:00 authored by Cesar A. Lozano HuertaLet $X$ 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

2014-06-20## Usage metrics

## Categories

No categories selected## Keywords

## Licence

## Exports

RefWorks

BibTeX

Ref. manager

Endnote

DataCite

NLM

DC