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Threefolds of degree 9 and 10 in \({\mathbb{P}}^ 5\). (English) Zbl 0723.14033
Let X be a smooth connected variety of dimension \(n\geq 2\) with canonical bundle \(K_ X\). Let L be a very ample line bundle on X. If the global sections of \(K_ X+(n-1)L\) span it, then they determine a map \(\Phi: X\to {\mathbb{P}}^ m\), called the adjunction map. A detailed study of adjunction maps has been done by A. J. Sommese (and also others) in a series of papers [e.g. in Complex analysis and algebraic geometry, Proc. Conf., Göttingen 1985, Lect. Notes Math. 1194, 175-213 (1986; Zbl 0601.14029)].
This paper carries it further \((n=3)\) to complete the classification of smooth threefolds X of degree 9 and 10 in \({\mathbb{P}}^ 5\). For such an X, it is shown that dim(\(\Phi\) (X))\(\geq 2\). If \(\dim(\Phi(X))\neq 3\), then X is a \({\mathbb{P}}^ 1\)-bundle over a minimal \(K_ 3\) surface, degree\((X)=9\), For \(\dim(\Phi(X))=3\), the classification uses minimal reduction \((X_ 0,L_ 0)\) of (X,L); \((X_ 0,L_ 0)\) is shown to be either of log general type or a conic bundle over a surface. For each class, examples are constructed using liaison. They turn out to be unique examples with given invariants.
The classification of smooth threefolds of degree \(\leq 8\) in \({\mathbb{P}}^ 5\) has been done by C. Okonek [Manuscr. Math. 38, 175-199 (1982; Zbl 0534.14023)] and Ionescu independently.

MSC:
14J30 \(3\)-folds
14J10 Families, moduli, classification: algebraic theory
14M06 Linkage
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[1] Arbarello, E., Cornalba, M., Griffiths, P.A., Harris, J.: Geometry of algebraic curves vol. I (Grundlehren, vol. 267). Berlin Heidelberg New York: Springer 1985 · Zbl 0559.14017
[2] Barth, W., Peters, C., Van de Ven, A.: Compact complex surfaces (Ergebnisse der Math. vol. 4). Berlin Heidelberg New York: Springer 1984
[3] Beauville, A.: Surfaces algébriques complexes. Astérisque54 (1978) · Zbl 0394.14014
[4] Beltrametti, M., Biancofiore, A., Sommese, A.J.: ProjectiveN-folds of log-general type I. Trans. Am. Math. Soc.314, 825-849 (1989) · Zbl 0702.14037
[5] Beltrametti, M., Sommese, A.J.: New properties of special varieties arising from the adjunction theory. preprint · Zbl 0754.14027
[6] Chang, M.C.: Classification of Buchsbaum subvarieties of codimension 2 in projective space. Crelle’s J.401, 101-112 (1989) · Zbl 0672.14026
[7] Fulton, W., Lazarsfeld, R.: Connectivity and its applications in algebraic geometry. Proceedings University of Illinois at Chicago Circle, 1980. (Lect. Notes Math. vol. 862). Berlin Heidelberg New York: Springer 1981 · Zbl 0484.14005
[8] Gruson, L., Peskine, C.: Genre des courbes de l’espace projectif. Algebraic geometry, Proceedings Tromsö, Norway 1977. (Lect. Notes Math. vol. 687) Berlin Heidelberg New York: Springer 1978 · Zbl 0412.14011
[9] Harris, J.: A bound on the geometric genus of projective varieties. Ann. Sc. Norm. Super. Pisa Cl. Sci., IV. Ser.8, 35-68 (1981) · Zbl 0467.14005
[10] Hartshorne, A.: Algebraic geometry. (G.T.M. vol. 52) Berlin Heidelberg New York: Springer 1977 · Zbl 0367.14001
[11] Ionescu, P.: Embedded projective varieties of small invariants. Proceedings of the Week of Algebraic Geometry, Bucharest 1982 (Lect. Notes Math. vol 1056) Berlin Heidelberg New York: Springer 1984
[12] Ionescu, P.: Generalized adjunction and applications, Math. Proc. Camb. Phil. Soc.99, 457-472 (1986) · Zbl 0619.14004 · doi:10.1017/S0305004100064409
[13] Ionescu, P.: Embedded projective varieties of small invariants, II. Rev. Roum. Math. Pures Appl.31, 539-544 (1986) · Zbl 0606.14038
[14] Ionescu, P.: Embedded projective varieties of small invariants, III. Preprint Series in Mathematics, n. 59/1988, Bucharest. · Zbl 0709.14023
[15] Iskovskih, V. A., Shokurov, V. V.: Biregular theory of Fano 3-folds, Proc. Alg. Geom., Copenhagen, 1978 (Lect. Notes Math. vol. 732, pp. 171-182), Berlin Heidelberg New York: Springer 1979
[16] Kawamata, Y.: The cone of curves of algebraic varieties. Ann. Math.119, 603-633 (1984) · Zbl 0544.14009 · doi:10.2307/2007087
[17] Livorni, E. L., Sommese, A. J.: Threefolds of non negative Kodaira dimension with sectional genus less than or equal to 15. Ann. Sc. Norm. Sup. Pisca. Cl. Sci. Ser. IV,13, 537-558 (1986) · Zbl 0636.14014
[18] Miyaoka, Y.: The Chern Classes and Kodaira Dimension of a Minimal Variety. Algebraic Geometry, Sendai 1985, 449-476. (Adv. Stud. Pure Math. vol. 10) Amsterdam: North Holland 1987
[19] Mori, S., Mukai, S.: Classification of Fano 3-folds withB 2-2, Manuscr. Math.36, 147-162 (1981) · Zbl 0478.14033 · doi:10.1007/BF01170131
[20] Mukai, S.: New classification of Fano threefolds and Fano manifolds of coidex 3, preprint (1988)
[21] Okonek, C.: 3-Mannigfaltigkeiten imP 5 und ihre zugehörigen stabilen Garben, Manuscr. Math.38, 175-199 (1982) · Zbl 0534.14023 · doi:10.1007/BF01168590
[22] Okonek, C.: Über 2-codimensionale Untermannigfaltigkeiten vom Grad 7 inP 4 undP 5. Math. Z.187, 209-219 (1984) · Zbl 0575.14030 · doi:10.1007/BF01161705
[23] Okonek, C.: On codimension-2 submanifolds inP 4 andP 5. Math. Gottingensis, Schriftenr. Sonderforschungsbereichs. Geom. Anal.50 (1986) · Zbl 0603.14035
[24] Okonek, C., Schneider, M., Spindler, H.: Vector bundles on complex projective spaces. Prog. Math.3 (1980) · Zbl 0438.32016
[25] Peskine, C.: Szpiro, L.: Liaison des variétés algébriques I. Invent. Math.26, 271-302 (1974) · Zbl 0298.14022 · doi:10.1007/BF01425554
[26] Ranestad, K.: On smooth surfaces of degree 10 in the projective fourspace. Preprint · Zbl 0827.14023
[27] Sommese, A. J.: Hyperplane sections of projective surfaces, I?The adjunction mapping. Duke Math. J.46, 377-401 (1979) · Zbl 0415.14019 · doi:10.1215/S0012-7094-79-04616-7
[28] Sommese, A. J.: Hyperplane sections, Proceedings of the Algebraic Geometry Conference, University of Illinois at Chicago Circle, 1980 (Lect. Notes Math. vol. 862, pp. 232-271). Berlin Heidelberg New York: Springer 1981
[29] Sommese, A. J.: Complex subspaces of homogeneous complex manifolds?II: Homotopy results. Nagoya Math. J.86, 101-129 (1982) · Zbl 0497.32026
[30] Sommese, A. J.: Ample divisors on Gorenstein varieties. Proceedings of Complex Geometry Conference, Nancy 1985. Rev. Inst. Cartan10, 104-125 (1986) · Zbl 0646.14007
[31] Sommese, A. J.: On the adjunction theoretic structure of projective varieties. Complex Analysis and Algebraic Geometry, Proceedings Göttingen 1985, (Lect. Notes Math. vol. 1194 pp. 175-213.) (1986) Berlin Heidelberg
[32] Sommese, A. J.: On the nonemptiness of the adjoint linear system of a hyperplane section of a threefold. Crelle’s J.402, 211-220 (1989) · Zbl 0675.14005
[33] Sommese, A. J., Van de Ven, A.: On the adjunction mapping. Math. Ann.278, 593-603 (1987) · Zbl 0655.14001 · doi:10.1007/BF01458083
[34] Van de Ven, A.: On the 2-connectedness of very ample divisors on a surface. Duke Math. J.,46, 403-407 (1979) · Zbl 0458.14003 · doi:10.1215/S0012-7094-79-04617-9
[35] Wilson, P. M. H.: Towards birational classification of algebraic varieties. Bull. Lond. Math. Soc.19, 1-48 (1987) · Zbl 0612.14033 · doi:10.1112/blms/19.1.1
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