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Bounding the Castelnuovo-Mumford Regularity of Algebraic Varieties

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posted on 14.12.2012, 00:00 by Wenbo Niu
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.

History

Advisor

Ein, Lawrence

Department

Department of Mathematics, Statistics, and Computer Science

Degree Grantor

University of Illinois at Chicago

Degree Level

Doctoral

Committee Member

Arapura, Donu Coskun, Izzet Popa, Mihnea Schnell, Christian

Submitted date

2011-08

Language

en

Issue date

14/12/2012