Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

On codimension two subvarieties of $\textbf {P\/}^5$ and $\textbf {P\/}^6$


Authors: Ph. Ellia and D. Franco
Journal: J. Algebraic Geom. 11 (2002), 513-533
DOI: https://doi.org/10.1090/S1056-3911-02-00320-X
Published electronically: March 21, 2002
MathSciNet review: 1894936
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Abstract | References | Additional Information

Abstract:

We prove the following:

Theorem. Let $X\subset \mathbf {P}^5$ be a smooth, subcanonical threefold. If $h^0(\mathcal {I}_X(4))\ne 0$, then $X$ is a complete intersection.

Let $X\subset \mathbf {P}^6$ be a smooth, codimension two subvariety, if $h^0(\mathcal {I}\!_X(5))\!\ne 0$ or $\operatorname {deg}(X)\le 73$, then $X$ is a complete intersection.

This improves, for $5\le n\le 6$, earlier results on Hartshorne’s conjecture for codimension two subvarieties of $\mathbf {P}^n$.


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

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Additional Information

Ph. Ellia
Affiliation: Dipartimento di Matematica, via Machiavelli, 35, 44100 Ferrara, Italy
Email: phe@dns.unife.it

D. Franco
Affiliation: Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Univ. Napoli “Federico II", Via Cintia, Monte S. Angelo 80126 Napoli, Italy
Email: dfranco@matna2.dma.unina.it

Received by editor(s): September 6, 1999
Published electronically: March 21, 2002
Additional Notes: Both authors are partially supported by MURST and Ferrara University in the framework of the project: “Geometria algebrica, algebra commutativa e aspetti computazionali"