Polarized surfaces of genus
Authors:
Maria Lucia Fania and Elvira Laura Livorni
Journal:
Trans. Amer. Math. Soc. 328 (1991), 445463
MSC:
Primary 14C20; Secondary 14D20, 14J25
MathSciNet review:
992607
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Abstract: Let be a smooth, complex, algebraic, projective surface and let be an ample line bundle on it. Let denote the genus of the pair . The purpose of this paper is to classify such pairs under the assumption that and the complete linear system contains a smooth curve. If and , Fujita has shown that is very ample and . If and , then and those pairs have been studied by Fujita and Beltrametti, Lanteri, and Palleschi. To study the remaining cases we have examined the two possibilities of being nef or not, for . In the cases in which is nef it turned out to be very useful to iterate the adjunction mapping for ample line bundles as it was done by Biancofiore and Livorni in the very ample case. If there are still open cases to solve in which completely different methods are needed.
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John Sommese, On the adjunction theoretic structure of projective
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 [BLP]
 M. Beltrametti, A. Lanteri, and M. Palleschi, Algebraic surfaces containing an ample divisor of arithmetic genus two, Ark. Mat. 25 (1987), 189210. MR 923406 (89e:14037)
 [BL]
 A. Biancofiore and E. L. Livorni, On the iteration of the adjunction process in the study of rational surfaces, Indiana Math. J. 36 (1987), 167187. MR 876997 (88d:14022)
 [BL]
 , On the iteration of the adjunction process for surfaces of negative Kodaira dimension, Manuscripta Math. 64 (1989), 3554. MR 994380 (90c:14004)
 [BL]
 , Algebraic ruled surfaces of low sectional genus, Ricerche Mat. 36 (1987), 1732. MR 977050 (89m:14021)
 [F]
 T. Fujita, On the structure of polarized varieties of genus zero, J. Fac. Sci. Univ. Tokyo 22 (1975), 103115. MR 0369363 (51:5596)
 [F]
 , On polarized varieties of small genera, Tôhoku Math. J. 34 (1982), 319341. MR 676113 (85e:14013)
 [F]
 , On hyperelliptic polarized varieties, Tôhoku Math. J. 35 (1983), 144. MR 695657 (84h:14016)
 [F]
 , On polarized manifolds of genus two, J. Math. Soc. Japan 36 (1984), 709730. MR 759426 (85m:14015)
 [F]
 , On polarized manifolds whose adjoint bundles are not semipositive, Advanced Study in Pure Math., vol. 10, NorthHolland, Amsterdam, 1987, pp. 167178. MR 946238 (89d:14006)
 [F]
 , Classification of polarized manifolds of sectional genus two, preprint.
 [F]
 , On classification of polarized manifolds by sectional genus, preprint.
 [F]
 , Polarized manifolds of degree three and genus two, J. Math. Soc. Japan 41 (1989), 311331. MR 984755 (90e:14037)
 [F]
 , On the structure of polarized manifolds with total deficiency one. III, J. Math. Soc. Japan 36 (1984), 7589. MR 723595 (85e:14061)
 [H]
 R. Hartshorne, Algebraic geometry, Graduate Texts in Math., no. 52, Springer, 1977. MR 0463157 (57:3116)
 [I]
 P. Ionescu, Ample and very ample divisors on surfaces, Rev. Roumaine Math. Pures Appl. 33 (1988), 349358. MR 950131 (89f:14037)
 [La]
 A. Lanteri, On polarized surfaces of genus two, Ann. Mat. Pura Appl. (4) 151 (1988), 317329. MR 964517 (90a:14053)
 [LP]
 A. Lanteri and M. Palleschi, About the adjunction process for polarized algebraic surfaces, J. Reine Angew. Math. 352 (1984), 1523. MR 758692 (86h:14028)
 [Li]
 E. L. Livorni, Classification of algebraic surfaces with sectional genus less than or equal to six. III: Ruled surfaces with , Math. Scand. 59 (1986), 929. MR 873485 (88d:14024)
 [Li]
 , Classification of algebraic surfaces with sectional genus less than or equal to six. II: Ruled surfaces with , Canad. J. Math. 38 (1986), 11101121. MR 869716 (88d:14023)
 [S]
 A. J. Sommese, Hyperplane sections of projective surfaces I. The adjunction mapping, Duke Math. J. 46 (1979), 377401. MR 534057 (82f:14033)
 [S]
 , On the birational theory of hyperplane sections of projective threefolds, unpublished 1981 manuscript.
 [S]
 , On the adjunction theoretic structure of projective varieties, Proc. Conf. Complex Analysis and Algebraic Geometry (Gottingen, 1985), (H. Grauert, ed.), Lecture Notes in Math., vol. 1194, Springer, 1986. MR 855885 (87m:14049)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199109926077
PII:
S 00029947(1991)09926077
Keywords:
Surfaces,
genus,
sectional genus
Article copyright:
© Copyright 1991 American Mathematical Society
