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Regularity, partial elimination ideals and the canonical bundle

Author: Matthew G. Jones
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 1531-1541
MSC (2000): Primary 51N15
Published electronically: December 23, 2003
MathSciNet review: 2053362
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Abstract: We present partial elimination ideals, which set-theoretically cut out the multiple point loci of a generic projection of a projective variety, as a way to bound the regularity of a variety in projective space. To do this, we utilize a combination of initial ideal methods and geometric methods. We first define partial elimination ideals and establish through initial ideal methods the way in which, for a given ideal, the regularity of the partial elimination ideals bounds the regularity of the given ideal. Then we explore the partial elimination ideals as a way to compute the canonical bundle of the generic projection of a variety and the canonical bundles of the multiple point loci of the projection, and we use Kodaira Vanishing to bound the regularity of the partial elimination ideals.

References [Enhancements On Off] (What's this?)

  • [BS87] David Bayer and Michael Stillman, A criterion for detecting $m$-regularity, Invent. Math. 87 (1987), no. 1, 1-11. MR 87k:13019
  • [EG84] David Eisenbud and Shiro Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), no. 1, 89-133. MR 85f:13023
  • [GLP83] L. Gruson, R. Lazarsfeld, and C. Peskine, On a theorem of Castelnuovo, and the equations defining space curves, Invent. Math. 72 (1983), no. 3, 491-506. MR 85g:14033
  • [Gre98] Mark Green, Generic initial ideals, Six Lectures On Commutative Algebra (Elias, 1996) (J. Elias, J. M. Giral, R. M. Miró-Roig, and S. Zarzuela, eds.), Birkhäuser-Verlag, Basel, 1998, Papers from the Summer School on Commutative Algebra held in Bellaterra, July 16-26, 1996, pp. 119-186. MR 99m:13040
  • [Kle82] Steven L. Kleiman, Multiple point formulas for maps, Enumerative geometry and classical algebraic geometry (Nice, 1981), Birkhäuser Boston, Mass., 1982, pp. 237-252. MR 84j:14059
  • [Kwa98] Sijong Kwak, Castelnuovo regularity for smooth subvarieties of dimensions $3$ and $4$, J. Algebraic Geom. 7 (1998), no. 1, 195-206. MR 2000d:14043
  • [Laz87] Robert Lazarsfeld, A sharp Castelnuovo bound for smooth surfaces, Duke Math. J. 55 (1987), no. 2, 423-429. MR 89d:14007
  • [Pin86] Henry C. Pinkham, A Castelnuovo bound for smooth surfaces, Invent. Math. 83 (1986), no. 2, 321-332. MR 87c:14044
  • [Xu94] Geng Xu, Subvarieties of general hypersurfaces in projective space, J. Differential Geom. 39 (1994), no. 1, 139-172. MR 95d:14043

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Additional Information

Matthew G. Jones
Affiliation: Department of Mathematics, California State University, Dominguez Hills, 1000 E. Victoria St., Carson, California 90747

Received by editor(s): October 5, 2001
Received by editor(s) in revised form: January 22, 2003
Published electronically: December 23, 2003
Additional Notes: The bulk of this work was completed under the direction of Mark Green as part of my Ph.D. thesis at UCLA. I am very grateful to Professor Green for all his guidance, the time and energy he devoted to me and the knowledge he imparted upon me. I am also grateful to the UCLA Mathematics Department for its support
Communicated by: Michael Stillman
Article copyright: © Copyright 2003 American Mathematical Society

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