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 $\mathbf {P}^n$.
BallCh 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)
Barth Barth, W.: “Transplanting cohomology classes in complex-projective space", Amer. J. Math., 92, 951-967 (1970)
BVdV Barth, W.; Van de Ven, A.: “On the geometry in codimension 2 of Grassmann manifolds", in L.N.M., 412, 1-35 (1974)
Bar Barth, W.: “Submanifolds of low codimension in projective space", Proc. of Int. Congr. of Math., Vancouver, 409-413 (1974)
EP Ellingsrud, G.; Peskine, Ch.: “Sur les surfaces lisses de $\textbf {P\/}^4$", Invent. Math., 95, 1-11 (1989)
Flo Fløystad, G.: “Curves on normal surfaces" (preprint)
Fu Fulton, W.: “Intersection theory", Erg. der Math. 3. Folge, 2, Springer (1984)
GP Gruson, L.; Peskine, Ch.: “Genre des courbes de l’espace projectif", in L.N.M. 687, 31-59 (1978)
Ha Hartshorne, R.: “Varieties of small codimension in projective space", Bull. A.M.S., 80, 1017-1032 (1974)
Ha2 Hartshorne, R.: “Complete intersections and connectedness", Amer. J. Math., 84, 497-508 (1962)
Ha3 Hartshorne, R.: “Generalized divisors on Gorenstein curves and a theorem of Noether", J. Math. Kyoto Univ., 26, 375-386 (1986)
Ho Holme, A.: “Codimension 2 subvarieties of projective space", Manuscripta Math., 65, 427-446 (1989)
HS Holme, A.; Schneider, M.: “A computer aided approach to codimension $2$ subvarieties of $\textbf {P\/}^n,n \geq 6$", J. Reine Angew. Math. (Crelle’s J.), 357, 205-220 (1985)
Koe Koelblen, L.: “Surfaces de $\textbf {P\/}^4$ tracées sur une hypersurface cubique", J. Reine Angew. Math. (Crelle’s J.), 433, 113-141 (1992)
LVdV 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).
Ma Manolache, N.: “Nilpotent lci structures on global complete intersections", Math. Z., 229, 403-411 (1995)
Ran Ran, Z.: “On projective varieties of codimension 2", Invent. Math., 73, 333-336 (1983)
Serre Serre, J.P.: “Groupes algébriques et corps de classes", Hermann Ed., (1975)
Zak Zak, F.L.: “Structure of Gauss maps", Funct. Anal. Appl., 21, 32-41 (1987)
Zak2 Zak, F.L.: “Projections of algebraic varieties", Math. USSR, Sb., 44, 535-544 (1983)
BallCh 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)
Barth Barth, W.: “Transplanting cohomology classes in complex-projective space", Amer. J. Math., 92, 951-967 (1970)
BVdV Barth, W.; Van de Ven, A.: “On the geometry in codimension 2 of Grassmann manifolds", in L.N.M., 412, 1-35 (1974)
Bar Barth, W.: “Submanifolds of low codimension in projective space", Proc. of Int. Congr. of Math., Vancouver, 409-413 (1974)
EP Ellingsrud, G.; Peskine, Ch.: “Sur les surfaces lisses de $\textbf {P\/}^4$", Invent. Math., 95, 1-11 (1989)
Flo Fløystad, G.: “Curves on normal surfaces" (preprint)
Fu Fulton, W.: “Intersection theory", Erg. der Math. 3. Folge, 2, Springer (1984)
GP Gruson, L.; Peskine, Ch.: “Genre des courbes de l’espace projectif", in L.N.M. 687, 31-59 (1978)
Ha Hartshorne, R.: “Varieties of small codimension in projective space", Bull. A.M.S., 80, 1017-1032 (1974)
Ha2 Hartshorne, R.: “Complete intersections and connectedness", Amer. J. Math., 84, 497-508 (1962)
Ha3 Hartshorne, R.: “Generalized divisors on Gorenstein curves and a theorem of Noether", J. Math. Kyoto Univ., 26, 375-386 (1986)
Ho Holme, A.: “Codimension 2 subvarieties of projective space", Manuscripta Math., 65, 427-446 (1989)
HS Holme, A.; Schneider, M.: “A computer aided approach to codimension $2$ subvarieties of $\textbf {P\/}^n,n \geq 6$", J. Reine Angew. Math. (Crelle’s J.), 357, 205-220 (1985)
Koe Koelblen, L.: “Surfaces de $\textbf {P\/}^4$ tracées sur une hypersurface cubique", J. Reine Angew. Math. (Crelle’s J.), 433, 113-141 (1992)
LVdV 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).
Ma Manolache, N.: “Nilpotent lci structures on global complete intersections", Math. Z., 229, 403-411 (1995)
Ran Ran, Z.: “On projective varieties of codimension 2", Invent. Math., 73, 333-336 (1983)
Serre Serre, J.P.: “Groupes algébriques et corps de classes", Hermann Ed., (1975)
Zak Zak, F.L.: “Structure of Gauss maps", Funct. Anal. Appl., 21, 32-41 (1987)
Zak2 Zak, F.L.: “Projections of algebraic varieties", Math. USSR, Sb., 44, 535-544 (1983)
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"
