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University of Illinois – Chicago

1. Niu, Wenbo. Bounding the Castelnuovo-Mumford Regularity of Algebraic Varieties.

Degree: 2012, University of Illinois – Chicago

URL: http://hdl.handle.net/10027/9630

In this monograph, we study bounds for the Castelnuovo-Mumford regularity of algebraic varieties.
In chapter three, we give a computational bounds for an homogeneous ideal, which extend a result of Chardin and Ulrich. Our approach is based on liaison theory and a study on singularities in a generic linkage.
In chapter four, via Nadel's vanishing theorems and multiplier ideal sheaves, we obtain a vanishing theorem for an ideal sheaf, which extends a result of Bertram, Ein and Lazarsfeld and a result of deFernex and Ein. Our theorem also leads to a regularity bound for powers of ideal sheaves. We also discuss applications of multiplier ideal sheaves in the study of multiregularity on a biprojective space.
In Chapter five, we study the asymptotic behavior of the regularity of ideal sheaves, We showed that the asymptotic regularity can be bounded by linear functions, this answers a question raised by Cutkosky and Kurano, and also extends a result of Cutkosky, Ein and Lazarsfeld. We also study asymptotic regularity of symbolic powers and give liner function bounds under some conditions.
In Chapter six, we give a sharp regularity bounds for a normal surface with rational, Gorenstein elliptic, log canonical singularities. This result verifies a conjecture of Eisenbud-Goto in normal surfaces case.
In Chapter seven, we study a notion of Mukai regularity on abelian varieties. We give a bound for M-regularity of curves in abelian varieties. Our approach is based on vanishing theorems and multiplier ideal sheaves.
*Advisors/Committee Members: Ein, Lawrence (advisor), Arapura, Donu (committee member), Coskun, Izzet (committee member), Popa, Mihnea (committee member), Schnell, Christian (committee member).*

Subjects/Keywords: Castelnuovo-Mumford regularity; powers of ideals; symbolic powers; multiplier ideal sheaves; vanishing theorems; asymptotic regularity; multiregularity; Mukai regularity.

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

APA (6^{th} Edition):

Niu, W. (2012). Bounding the Castelnuovo-Mumford Regularity of Algebraic Varieties. (Thesis). University of Illinois – Chicago. Retrieved from http://hdl.handle.net/10027/9630

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):

Niu, Wenbo. “Bounding the Castelnuovo-Mumford Regularity of Algebraic Varieties.” 2012. Thesis, University of Illinois – Chicago. Accessed July 12, 2020. http://hdl.handle.net/10027/9630.

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

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Niu, Wenbo. “Bounding the Castelnuovo-Mumford Regularity of Algebraic Varieties.” 2012. Web. 12 Jul 2020.

Vancouver:

Niu W. Bounding the Castelnuovo-Mumford Regularity of Algebraic Varieties. [Internet] [Thesis]. University of Illinois – Chicago; 2012. [cited 2020 Jul 12]. Available from: http://hdl.handle.net/10027/9630.

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

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Niu W. Bounding the Castelnuovo-Mumford Regularity of Algebraic Varieties. [Thesis]. University of Illinois – Chicago; 2012. Available from: http://hdl.handle.net/10027/9630

Not specified: Masters Thesis or Doctoral Dissertation

2. Walker, Robert. Uniform Symbolic Topologies in Non-Regular Rings.

Degree: PhD, Mathematics, 2019, University of Michigan

URL: http://hdl.handle.net/2027.42/149907

When does a Noetherian commutative ring R have uniform symbolic topologies (USTP) on primes – read, when does there exist an integer D>0 such that the symbolic power P^{(Dr)} lies in P^{r} for all prime ideals P in R and all r >0? Groundbreaking work of Ein – Lazarsfeld – Smith, as extended by Hochster and Huneke, and by Ma and Schwede in turn, provides a beautiful answer in the setting of finite-dimensional excellent regular rings. Their work shows that there exists a D depending only on the Krull dimension: in other words, the exact same D works for all regular rings as stated of a fixed dimension.
Referring to this last observation, we say in the thesis that the class of excellent regular rings enjoys class solidarity relative to the uniform symbolic topology property (USTP class solidarity), a strong form of uniformity. In contrast, this thesis shows that for certain classes of non-regular rings including rational surface singularities and select normal toric rings, a uniform bound D does exist but depends on the ring, not just its dimension. In particular, for rational double point surface singularities over the field C of complex numbers, we show that USTP solidarity is plainly impossible.
It is natural to sleuth for analogues of the Improved Ein – Lazarsfeld – Smith Theorem where the ring R is non-regular, or where the above ideal containments can be improved using a linear function whose growth rate is slower. This thesis lies in the overlap of these research directions, working with Noetherian domains.
*Advisors/Committee Members: Smith, Karen E (committee member), Jacobson, Daniel (committee member), Hochster, Mel (committee member), Jeffries, Jack (committee member), Koch, Sarah Colleen (committee member), Speyer, David E (committee member).*

Subjects/Keywords: Symbolic Powers of Ideals in Noetherian Integral Domains; Rationally Singular Combinatorially Defined Algebras; Weil divisor class groups of Noetherian normal integral domains; Mathematics; Science

…*powers* *of*
*ideals* in Noetherian commutative rings, relative to regular *powers*. Echoing Sarah… …We investigate two collections *of* *ideals*, namely, the regular and symbolic *powers* *of*
a… …the symbolic
*powers* *of* I are a family *of* *ideals* {I (N ) } in R indexed… …prime *ideals* are maximal, and one can show that (symbolic) *powers* *of* distinct… …subvarieties *of* V are in bijection
with prime *ideals*. In particular, the points p in V correspond to…

Record Details Similar Records

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

APA (6^{th} Edition):

Walker, R. (2019). Uniform Symbolic Topologies in Non-Regular Rings. (Doctoral Dissertation). University of Michigan. Retrieved from http://hdl.handle.net/2027.42/149907

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

Walker, Robert. “Uniform Symbolic Topologies in Non-Regular Rings.” 2019. Doctoral Dissertation, University of Michigan. Accessed July 12, 2020. http://hdl.handle.net/2027.42/149907.

MLA Handbook (7^{th} Edition):

Walker, Robert. “Uniform Symbolic Topologies in Non-Regular Rings.” 2019. Web. 12 Jul 2020.

Vancouver:

Walker R. Uniform Symbolic Topologies in Non-Regular Rings. [Internet] [Doctoral dissertation]. University of Michigan; 2019. [cited 2020 Jul 12]. Available from: http://hdl.handle.net/2027.42/149907.

Council of Science Editors:

Walker R. Uniform Symbolic Topologies in Non-Regular Rings. [Doctoral Dissertation]. University of Michigan; 2019. Available from: http://hdl.handle.net/2027.42/149907