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On smooth subcanonical varieties of codimension 2 in \(P^ n\), n\(\geq 4\). (English) Zbl 0549.14015
Let \(X\subset {\mathbb{P}}_ N({\mathbb{C}})\) be a codimension 2 smooth submanifold with \(\omega_ X\cong {\mathcal O}_ X(e),\) \(e\leq 0\). Here we prove that X is a complete intersection, proving that it has the degree of a complete intersection and then applying a recent result of Z. Ran [Invent. Math. 73, 333-336 (1983; Zbl 0521.14018)]. If \(e<0\) the assertion about the degree was proved independently, simultaneously and in the same way by Y. Sakane in Saitama Math. J. 1, 9-27 (1983; Zbl 0544.14031).

MSC:
14M07 Low codimension problems in algebraic geometry
14M10 Complete intersections
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[1] Atiyah, M. F.; Reks, E., Vector Bundles on Projective 3-space, Iuv. Math., 35, 131-153 (1976) · Zbl 0332.32020
[2] Barth, W., Some properties of stable rank 2vector bundles on P_n, Math. Ann., 226, 125-150 (1977) · Zbl 0332.32021
[3] W.Barth - G.Elencwajg,Concernant la cohomologie des fibres algebriques stable sur Pn(C), P. 1-24 on:Varietes Analitiques Compactes — Colloque Nice 1977, Springer Lecture Notes n. 683 (1978).
[4] W.Barth - A.Van De Ven,On the geometry in codimension 2of Grassmann manifolds, P. 1-35 on:Classification of Algebraic Varieties and Compact Complex Manifolds, Springer Lecture Notes, n. 412 (1974).
[5] A.Beauville,Surfaces Algebriques Complexes, Asterisque, n. 54 (1978). · Zbl 0394.14014
[6] Ciliberto, C., Canonical surfaces withp_g=p_a=5and K^2=10, Annali Scuola Norm. Sup. Pisa, serie IV, 9, 287-336 (1982)
[7] Elencwajg, G., Les Fibres Uniformes de rang 3sur P_2(C) sont Homogenes, Math. Ann., 231, 217-227 (1978) · Zbl 0378.14003
[8] Elencwajg, G.; Foster, D., Bounding cohomology groups of vector bundles on P_n, Math. Ann., 246, 251-270 (1980) · Zbl 0432.14011
[9] Evans, E. G.; Griffith, P., The syzygy problem, Ann. of Math., 214, n. 2, 323-333 (1981) · Zbl 0497.13013
[10] W.Fulton - R.Lazarsfeld,Connectivity and its applications in Algebraic Geometry, on:Algebraic Geometry, Proceedings, University of Illinois at Chicago Circle, Springer Lecture Notes, n. 862 (1980). · Zbl 0484.14005
[11] Gherardelli, G., Sulle curve sghembe algebriche intersezioni complete di due superficie, Atti dell’Accademia Reale d’Italia, XXI, 128-132 (1942) · Zbl 0061.35802
[12] Grauert, H.; Schneider, M., Komplexe UnterrÄume und holomorphe Vectorraumbundel vom Rang zwei, Math. Ann., 230, 75-90 (1977) · Zbl 0412.32014
[13] L.Gruson - C.Peskine,Genre des courbes dans l’espace projectif, P. 31-59, on:Algebraic Geometry Proceedings Tromso 1977, Springer Lecture Notes, n. 687 (1978).
[14] Hartshorne, R., Algebraic Geometry (1977), Berlin Heidelberg, New York: Springer, Berlin Heidelberg, New York
[15] Hartshorne, R., Varieties of small codimension in projective space, Bull. A.M.S., 80, 1017-1032 (1974) · Zbl 0304.14005
[16] Hartshorne, R., Stable vector bundles onP^3, Math. Ann., 238, 229-280 (1978) · Zbl 0411.14002
[17] R.Hartshorne,Residues and Duality, Springer Lecture Notes, n. 20 (1971).
[18] Horrocks, G.; Mumford, D., A rank 2vector bundle onP^4with 15,000symmetries, Topology, 12, 63-81 (1973) · Zbl 0255.14017
[19] Maruyama, M., Boundedness of semi-stable sheaves of small ranks, Nagoya Math. J., 78, 65-94 (1980) · Zbl 0456.14011
[20] J. P.Murre,Classification of Fano threefolds according to Fano and Iskovski, on:Algebraic Threefolds, Springer Lecture Notes, n. 947 (1982). · Zbl 0492.14025
[21] Z.Ran,The class of an Hilbert scheme inside another, with applications to Projective Geometry and special divisors, (preprint).
[22] Rao, P., Liaison among curves inP^3, Inv. Math., 50, 205-217 (1979) · Zbl 0406.14033
[23] M.Schneider,Holomorphic vector bundles onP^n, P. 80-102, on:Seminaire Bourbaki 1978, Springer Lecture Notes, n. 770 (1978).
[24] Severi, F., Intorno ai punti doppi impropri di una superficie generale dello spazio a quattro dimensioni e ai suoi punti tripli apparenti, Rend. Circ. Mat. Palermo, 15, 33-51 (1901) · JFM 32.0648.04
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