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1. Lozano Huerta, Cesar A. Birational Geometry of the Space of Complete Quadrics.

Degree: 2014, University of Illinois – Chicago

URL: http://hdl.handle.net/10027/18779

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):= {Proj}(\bigoplus_{m ≥ 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 {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.
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Chapter 4 outlines the relation of this work with that of Semple , as well as future directions of research.
*Advisors/Committee Members: Coskun, Izzet (advisor), Ein, Lawrence (committee member), Popa, Mihnea (committee member), Huizenga, Jack (committee member), De Fernex, Tommaso (committee member).*

Subjects/Keywords: algebraic gemeotry; birational geometry; complete quadrics; minimal model program; Mori's program; Hassett-Keel program; moduli spaces

…rational) from
X to another projective *algebraic* variety. We will do so, following the… …those Q-factorial *algebraic* varieties for which N 1 (X) ⌦ Q = Pic(X) ⌦ Q…

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APA (6^{th} Edition):

Lozano Huerta, C. A. (2014). Birational Geometry of the Space of Complete Quadrics. (Thesis). University of Illinois – Chicago. Retrieved from http://hdl.handle.net/10027/18779

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Lozano Huerta, Cesar A. “Birational Geometry of the Space of Complete Quadrics.” 2014. Thesis, University of Illinois – Chicago. Accessed July 12, 2020. http://hdl.handle.net/10027/18779.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Lozano Huerta, Cesar A. “Birational Geometry of the Space of Complete Quadrics.” 2014. Web. 12 Jul 2020.

Vancouver:

Lozano Huerta CA. Birational Geometry of the Space of Complete Quadrics. [Internet] [Thesis]. University of Illinois – Chicago; 2014. [cited 2020 Jul 12]. Available from: http://hdl.handle.net/10027/18779.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Lozano Huerta CA. Birational Geometry of the Space of Complete Quadrics. [Thesis]. University of Illinois – Chicago; 2014. Available from: http://hdl.handle.net/10027/18779

Not specified: Masters Thesis or Doctoral Dissertation