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University of Illinois – Chicago

1. Adali, Riza Seckin. Singular Loci of Restriction Varieties.

Degree: 2016, University of Illinois – Chicago

URL: http://hdl.handle.net/10027/21300

Restriction varieties in the orthogonal Grassmannian are subvarieties of OG(k, n) defined by rank conditions given by a flag that is not necessarily isotropic with respect to the relevant symmetric bilinear form. In particular orthogonal Schubert varieties are examples of restriction varieties. In this thesis, we describe a resolution of singularities for restriction varieties in OG(k, n). We study the relationship between the exceptional locus of the resolution and the singularities of a restriction variety. Our study results in a partial description of the singular locus of a restriction variety in terms of the components of the exceptional locus of the resolution of singularities.
*Advisors/Committee Members: Coskun, Izzet (advisor), Ein, Lawrence (committee member), Huizenga, Jack (committee member), Riedl, Eric (committee member), Tucker, Kevin (committee member).*

Subjects/Keywords: Restriction varieties; resolution of singularities; singular locus

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Adali, R. S. (2016). Singular Loci of Restriction Varieties. (Thesis). University of Illinois – Chicago. Retrieved from http://hdl.handle.net/10027/21300

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):

Adali, Riza Seckin. “Singular Loci of Restriction Varieties.” 2016. Thesis, University of Illinois – Chicago. Accessed July 12, 2020. http://hdl.handle.net/10027/21300.

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

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Adali, Riza Seckin. “Singular Loci of Restriction Varieties.” 2016. Web. 12 Jul 2020.

Vancouver:

Adali RS. Singular Loci of Restriction Varieties. [Internet] [Thesis]. University of Illinois – Chicago; 2016. [cited 2020 Jul 12]. Available from: http://hdl.handle.net/10027/21300.

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

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Adali RS. Singular Loci of Restriction Varieties. [Thesis]. University of Illinois – Chicago; 2016. Available from: http://hdl.handle.net/10027/21300

Not specified: Masters Thesis or Doctoral Dissertation

University of Illinois – Chicago

2. Ryan, Timothy L. The Effective Cone of Moduli Spaces of Sheaves on a Smooth Quadric Surface.

Degree: 2016, University of Illinois – Chicago

URL: http://hdl.handle.net/10027/21355

In this paper, we provide an approach to computing the effective cone of moduli spaces of sheaves on a smooth quadric surface. We find Brill-Noether divisors spanning extremal rays of the effective cone using resolutions of the general elements of the moduli space which are found using the machinery of exceptional bundles. We use this approach to provide many examples of extremal rays in these effective cones. In particular, we completely compute the effective cone of the first fifteen Hilbert schemes of points on a smooth quadric surface.
*Advisors/Committee Members: Coskun, Izzet (advisor), Ein, Lawrence (committee member), Huizenga, Jack (committee member), Riedl, Eric (committee member), Tucker, Kevin (committee member).*

Subjects/Keywords: algebraic geometry; moduli spaces; bridgeland stability; stability; birational geometry; effective cone; quadric surface; mmp; minimal model program

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Ryan, T. L. (2016). The Effective Cone of Moduli Spaces of Sheaves on a Smooth Quadric Surface. (Thesis). University of Illinois – Chicago. Retrieved from http://hdl.handle.net/10027/21355

Not specified: Masters Thesis or Doctoral Dissertation

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

Ryan, Timothy L. “The Effective Cone of Moduli Spaces of Sheaves on a Smooth Quadric Surface.” 2016. Thesis, University of Illinois – Chicago. Accessed July 12, 2020. http://hdl.handle.net/10027/21355.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Ryan, Timothy L. “The Effective Cone of Moduli Spaces of Sheaves on a Smooth Quadric Surface.” 2016. Web. 12 Jul 2020.

Vancouver:

Ryan TL. The Effective Cone of Moduli Spaces of Sheaves on a Smooth Quadric Surface. [Internet] [Thesis]. University of Illinois – Chicago; 2016. [cited 2020 Jul 12]. Available from: http://hdl.handle.net/10027/21355.

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Ryan TL. The Effective Cone of Moduli Spaces of Sheaves on a Smooth Quadric Surface. [Thesis]. University of Illinois – Chicago; 2016. Available from: http://hdl.handle.net/10027/21355

Not specified: Masters Thesis or Doctoral Dissertation

3. 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.
\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):= {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.
\medskip
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

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

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

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.

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.

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