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

Full-text PDF Free Access

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 𝑚-regularity*, Invent. Math.**87**(1987), no. 1, 1–11. MR**862710**, 10.1007/BF01389151**[EG84]**David Eisenbud and Shiro Goto,*Linear free resolutions and minimal multiplicity*, J. Algebra**88**(1984), no. 1, 89–133. MR**741934**, 10.1016/0021-8693(84)90092-9**[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**704401**, 10.1007/BF01398398**[Gre98]**Mark L. Green,*Generic initial ideals*, Six lectures on commutative algebra (Bellaterra, 1996) Progr. Math., vol. 166, Birkhäuser, Basel, 1998, pp. 119–186. MR**1648665****[Kle82]**Steven L. Kleiman,*Multiple point formulas for maps*, Enumerative geometry and classical algebraic geometry (Nice, 1981), Progr. Math., vol. 24, Birkhäuser, Boston, Mass., 1982, pp. 237–252. MR**685771****[Kwa98]**Sijong Kwak,*Castelnuovo regularity for smooth subvarieties of dimensions 3 and 4*, J. Algebraic Geom.**7**(1998), no. 1, 195–206. MR**1620706****[Laz87]**Robert Lazarsfeld,*A sharp Castelnuovo bound for smooth surfaces*, Duke Math. J.**55**(1987), no. 2, 423–429. MR**894589**, 10.1215/S0012-7094-87-05523-2**[Pin86]**Henry C. Pinkham,*A Castelnuovo bound for smooth surfaces*, Invent. Math.**83**(1986), no. 2, 321–332. MR**818356**, 10.1007/BF01388966**[Xu94]**Geng Xu,*Subvarieties of general hypersurfaces in projective space*, J. Differential Geom.**39**(1994), no. 1, 139–172. MR**1258918**

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:
http://dx.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