Journal of Algebraic Geometry

Author(s): Ph. Ellia; D. Franco
Journal: J. Algebraic Geom. 11 (2002), 513-533.
Posted: March 21, 2002
Retrieve article in: PDF DVI PostScript

Theorem. Let $X\subset\mathbf{P}^5$ be a smooth, subcanonical threefold. If $h^0(\mathcal{I}_X(4))\ne0$ , then is a complete intersection.

Let $X\subset\mathbf{P}^6$ be a smooth, codimension two subvariety, if $h^0(\mathcal{I}_X(5))\ne0$ or $\operatorname{deg}(X)\le73$ , then is a complete intersection.

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

1.: Ballico, E.; Chiantini, L.: ``On smooth subcanonical varieties of codimension 2 in $\textbf{P\/}^n,n\geq 4$ ", Annali di Matematica Pura ed Applicata, vol. CXXXV, 99-118 (1983)
2.: Barth, W.: ``Transplanting cohomology classes in complex-projective space", Amer. J. Math., 92, 951-967 (1970)
3.: Barth, W.; Van de Ven, A.: ``On the geometry in codimension 2 of Grassmann manifolds", in L.N.M., 412, 1-35 (1974)
4.: Barth, W.: ``Submanifolds of low codimension in projective space", Proc. of Int. Congr. of Math., Vancouver, 409-413 (1974)
5.: Ellingsrud, G.; Peskine, Ch.: ``Sur les surfaces lisses de $\textbf{P\/}^4$ ", Invent. Math., 95, 1-11 (1989)
6.: Fløystad, G.: ``Curves on normal surfaces" (preprint)
7.: Fulton, W.: ``Intersection theory", Erg. der Math. 3. Folge, 2, Springer (1984)
8.: Gruson, L.; Peskine, Ch.: ``Genre des courbes de l'espace projectif", in L.N.M. 687, 31-59 (1978)
9.: Hartshorne, R.: ``Varieties of small codimension in projective space", Bull. A.M.S., 80, 1017-1032 (1974)
10.: Hartshorne, R.: ``Complete intersections and connectedness", Amer. J. Math., 84, 497-508 (1962)
11.: Hartshorne, R.: ``Generalized divisors on Gorenstein curves and a theorem of Noether", J. Math. Kyoto Univ., 26, 375-386 (1986)
12.: Holme, A.: ``Codimension 2 subvarieties of projective space", Manuscripta Math., 65, 427-446 (1989)
13.: Holme, A.; Schneider, M.: ``A computer aided approach to codimension subvarieties of $\textbf{P\/}^n,n \geq 6$ ", J. Reine Angew. Math. (Crelle's J.), 357, 205-220 (1985)
14.: Koelblen, L.: ``Surfaces de $\textbf{P\/}^4$ tracées sur une hypersurface cubique", J. Reine Angew. Math. (Crelle's J.), 433, 113-141 (1992)
15.: Lazarsfeld, R.; Van de Ven, A.: ``Topics in the geometry of projective space (recent work of F.L. Zak)", DMV Seminar, band 4, Birkhäuser Verlag (1984).
16.: Manolache, N.: ``Nilpotent lci structures on global complete intersections", Math. Z., 229, 403-411 (1995)
17.: Ran, Z.: ``On projective varieties of codimension 2", Invent. Math., 73, 333-336 (1983)
18.: Serre, J.P.: ``Groupes algébriques et corps de classes", Hermann Ed., (1975)
19.: Zak, F.L.: ``Structure of Gauss maps", Funct. Anal. Appl., 21, 32-41 (1987)
20.: Zak, F.L.: ``Projections of algebraic varieties", Math. USSR, Sb., 44, 535-544 (1983)

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

PII: S 1056-3911(02)00320-X
Received by editor(s): September 6, 1999
Posted: 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"

Previous issue \| Table of contents \| Next issue Recently posted articles \| Most recent issue \| All issues