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University of Illinois – Chicago

1. Chou, Chih-Chi. Singularities in Birational Geometry.

Degree: 2014, University of Illinois – Chicago

URL: http://hdl.handle.net/10027/19077

In this thesis we study singularities in birational geometry.
In the first part, we investigate log canonical singularities and its relation with rational singularities.
In the second part, we prove a vanishing theorem of log canonical pairs. In the last part, we consider
singularities of normal varieties.
*Advisors/Committee Members: Ein, Lawrence (advisor), Coskun, Izzet (committee member), Popa, Mihnea (committee member), Tucker, Kevin (committee member), Niu, Wenbo (committee member).*

Subjects/Keywords: Log canonical singularities; Rational singularities; Vanishing theorems.

Record Details Similar Records

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

APA (6^{th} Edition):

Chou, C. (2014). Singularities in Birational Geometry. (Thesis). University of Illinois – Chicago. Retrieved from http://hdl.handle.net/10027/19077

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):

Chou, Chih-Chi. “Singularities in Birational Geometry.” 2014. Thesis, University of Illinois – Chicago. Accessed July 12, 2020. http://hdl.handle.net/10027/19077.

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

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Chou, Chih-Chi. “Singularities in Birational Geometry.” 2014. Web. 12 Jul 2020.

Vancouver:

Chou C. Singularities in Birational Geometry. [Internet] [Thesis]. University of Illinois – Chicago; 2014. [cited 2020 Jul 12]. Available from: http://hdl.handle.net/10027/19077.

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

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Chou C. Singularities in Birational Geometry. [Thesis]. University of Illinois – Chicago; 2014. Available from: http://hdl.handle.net/10027/19077

Not specified: Masters Thesis or Doctoral Dissertation

University of Illinois – Chicago

2. Song, Lei. Rational Singularities of Brill-Noether Loci and Log Canonical Thresholds on Hilbert Schemes of Points.

Degree: 2014, University of Illinois – Chicago

URL: http://hdl.handle.net/10027/18980

It is well known in algebraic geometry that Hilbert and Picard functors are representable by Hilbert schemes {Hilb}(X) and Picard schemes {Pic}(X) respectively. The thesis studies singularities of certain spaces relating to these schemes. It primarily consists of two parts of independent interest.
In the first part (Chapter 3), we study the Brill-Noether locus W^{0}(X) of effective line bundles over a smooth projective variety X of arbitrary dimension; and we show that if a line bundle L is semi-regular, then W^{0}(X) has rational singularities at [L]. Since the semi-regularity holds automatically for all line bundles over a curve, we thereby recover a Kempf's theorem stating that all Brill-Noether loci W^{0}_{d}(C) have rational singularities for all smooth projective curve C of genus g and 1 ≤ d ≤ g-1. We also study the local ring \sshf{W^{0}(X), [L]} for such L. To show the condition of semi-regularity is not overly strong, we construct a family of examples from ruled surfaces, and make an analysis of one type of components of W^{0}_{sr}(X).
In the second part (Chapter 4), we study the Hilbert scheme of n-points on a quasi-projective smooth surface X. Specifically, we show that the universal family Z^{n} over {Hilb}^{n}(X) has non ℚ-Gorenstein, rational singularities, and its Samuel multiplicity can be described by a quadric in terms of the dimension of socle of zero-dimensional subscheme. In a different but closely related direction, we study the log canonical threshold c_{n} of the pair ({Hilb}^{n}(X), B^{n}), where X is the affine plane and B^{n} is the exceptional divisor of the Hilbert-Chow morphism, via two approaches. Using the Fulton-MacPherson compactification of configuration spaces and Haiman's work on the n! conjecture, we give a lower bound of c_{n}. On the other hand, by versal deformations of monomial ideals on the plane, we relate c_{n} to the log canonical threshold of the discriminant of a degree n polynomial in one variable.
*Advisors/Committee Members: Ein, Lawrence (advisor), Coskun, Izzet (committee member), Libgober, Anatoly (committee member), Popa, Mihnea (committee member), Niu, Wenbo (committee member).*

Subjects/Keywords: Brill-Noether loci; Semi-regular line bundles; Rational singularities; Hilbert scheme of points on a surface; Universal family; Log canonical threshold

Record Details Similar Records

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

APA (6^{th} Edition):

Song, L. (2014). Rational Singularities of Brill-Noether Loci and Log Canonical Thresholds on Hilbert Schemes of Points. (Thesis). University of Illinois – Chicago. Retrieved from http://hdl.handle.net/10027/18980

Not specified: Masters Thesis or Doctoral Dissertation

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

Song, Lei. “Rational Singularities of Brill-Noether Loci and Log Canonical Thresholds on Hilbert Schemes of Points.” 2014. Thesis, University of Illinois – Chicago. Accessed July 12, 2020. http://hdl.handle.net/10027/18980.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Song, Lei. “Rational Singularities of Brill-Noether Loci and Log Canonical Thresholds on Hilbert Schemes of Points.” 2014. Web. 12 Jul 2020.

Vancouver:

Song L. Rational Singularities of Brill-Noether Loci and Log Canonical Thresholds on Hilbert Schemes of Points. [Internet] [Thesis]. University of Illinois – Chicago; 2014. [cited 2020 Jul 12]. Available from: http://hdl.handle.net/10027/18980.

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Song L. Rational Singularities of Brill-Noether Loci and Log Canonical Thresholds on Hilbert Schemes of Points. [Thesis]. University of Illinois – Chicago; 2014. Available from: http://hdl.handle.net/10027/18980

Not specified: Masters Thesis or Doctoral Dissertation