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Download fileBounding the Castelnuovo-Mumford Regularity of Algebraic Varieties
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posted on 2012-12-14, 00:00 authored by Wenbo NiuIn 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.