Abstract
Let (X,L) be a quasi-polarized variety, i.e. X is a smooth projective variety over the complex numbers \(\mathbb{C}\) and L is a nef and big divisor on X. Then we conjecture that g(L) = q(X), whereg(L) is the sectional genus of L and \(q(X) = \dim H^1 (\mathcal{O}_X )\). In this paper, we treat the case \(\dim X = 2\). First we prove that this conjecture is true for \(\kappa (X) \leqslant 1\), and we classify (X,L) withg(L)=q(X), where \(\kappa (X)\) is the Kodaira dimension of X. Next we study some special cases of\(\kappa (X) = 2\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barth, W., Peters, C. and Van de Ven, A.: Compact complex surfaces, Ergeb. Math. Grenz geb. 4, SpringerVerlag, Berlin, 1984.
Beauville, A.: Surfaces algébriques complexes, Astérisque 54 (1978).
Beauville, A.: L'application canonique pour les surfaces de type général, Invent.Math. 55 (1979), 121-140.
Beauville, A.: L'inégalité pg ≥2q-4 pour les surfaces de type général, Bull. Soc. Math. Fr. 110 (1982), 343-346.
Beltrametti,M., Lanteri, A. and Palleschi,M.: Algebraic surfaces containing an ample divisor of arithmetic genus two, Ark. Mat. 25 (1987), 189-210.
Bombieri, E.: Canonical models of surfaces of general type, Publ. Math. Inst. Hautes. Etude. Sci. 42 (1972), 171-219.
Debarre, O.: Inégalités numériques pour les surfaces de type général, Bull. Soc. Math. Fr. 110 (1982), 319-346. Addendum, Bull. Soc. Math. Fr. 111 (1983), 301-302.
Fujita, T.: On Kähler fiber spaces over curves, J. Math. Soc. Japan 30 (1978), 779-794.
Fujita, T: On polarized manifolds whose adjoint bundles are not semipositive, in: Algebraic Geometry, Sendai, 1985, Advanced Studies in Pure Math. 10, NorthHolland, Amsterdam, 1987, pp. 167-178.
Fujita, T.: Remarks on quasipolarized varieties, Nagoya Math. J. 115 (1989), 105-123.
Fujita, T: Classification Theories of Polarized Varieties, London Math. Soc. Lecture Note Ser. 155. Cambridge University Press, 1990.
Fujita, T.: On certain polarized elliptic surfaces, Geometry of Complex Projective Varieties Cetruroiggo, Seminars and Conferences 9 Mediterranean Press, 1993, pp. 153-163.
Fukuma, Y.: Master Thesis (in Japanese), 1993.
Hartshorne, R.: Algebraic Geometry, Graduate Texts in Math. 52 SpringerVerlag, New York, 1977.
Iitaka, S.: Algebraic Geometry, Graduate Texts in Math. 76 SpringerVerlag, New York, 1982.
Kawamata, Y.: A generalization of KodairaRamanujam's vanising theorem, Math. Ann. 261 (1982), 43-46.
Kollár, J.: Higher direct images of dualizing sheaves I, Ann. of Math. 123 (1986), 11-42.
Lanteri, A.: Algebraic surfaces containing a smooth curve of genus q(S) as an ample divisor, Geom. Dedicata 17 (1984), 189-197.
Lanteri, A. and Palleschi, M.: About the adjunction process for polarized algebraic surfaces, J. Reine Angew. Math. 352 (1984), 15-23.
Mori, S.: Threefolds whose canonical bundles are not numerically effective, Ann. of Math. 116 (1982), 133-176.
Mumford, Abelian Varieties, Oxford University Press, 1970.
Sakai, F.:Ample Cartier divisors on normal surfaces, J. Reine Angew.Math. 366 (1986), 121-128.
Serrano, F.: On isotrivial fibred surfaces, preprint.
Serrano, F.: The Picard group of a quasibundle, Manuscripta Math. 73 (1991), 63-82.
Xiao, G.: Fibered algebraic surfaces with low slope, Math. Ann. 276 (1987), 449-466.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Fukuma, Y. A Lower Bound for the Sectional Genus of Quasi-Polarized Surfaces. Geometriae Dedicata 64, 229–251 (1997). https://doi.org/10.1023/A:1004939700290
Issue Date:
DOI: https://doi.org/10.1023/A:1004939700290