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

DOI:
https://doi.org/10.1090/S0002-9939-03-07389-1

Published electronically:
December 23, 2003

MathSciNet review:
2053362

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[BS87]**David Bayer and Michael Stillman,*A criterion for detecting -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 and*, 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**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
51N15

Retrieve articles in all journals with MSC (2000): 51N15

Additional Information

**Matthew G. Jones**

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

Email:
mjones@csudh.edu

DOI:
https://doi.org/10.1090/S0002-9939-03-07389-1

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