On sectional genus of quasi-polarized 3-folds
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Abstract:
Let $X$ be a smooth projective variety over $\mathbb {C}$ and $L$ a nef-big (resp. ample) divisor on $X$. Then $(X,L)$ is called a quasi-polarized (resp. polarized) manifold. Then we conjecture that $g(L)\geq q(X)$, where $g(L)$ is the sectional genus of $L$ and $q(X)=\operatorname {dim}H^{1}(\mathcal {O}_{X})$ is the irregularity of $X$. In general it is unknown whether this conjecture is true or not, even in the case of $\operatorname {dim}X=2$. For example, this conjecture is true if $\operatorname {dim}X=2$ and $\operatorname {dim}H^{0}(L)>0$. But it is unknown if $\operatorname {dim}X\geq 3$ and $\operatorname {dim}H^{0}(L)>0$. In this paper, we prove $g(L)\geq q(X)$ if $\operatorname {dim}X=3$ and $\operatorname {dim}H^{0}(L)\geq 2$. Furthermore we classify polarized manifolds $(X,L)$ with $\operatorname {dim}X=3$, $\operatorname {dim}H^{0}(L)\geq 3$, and $g(L)=q(X)$.References
- Lucian Bădescu, On ample divisors, Nagoya Math. J. 86 (1982), 155–171. MR 661223, DOI 10.1017/S0027763000019838
- Lucian Bădescu, On ample divisors. II, Proceedings of the Week of Algebraic Geometry (Bucharest, 1980) Teubner-Texte zur Mathematik, vol. 40, Teubner, Leipzig, 1981, pp. 12–32. MR 712513
- Lucian Bădescu, The projective plane blown up at a point, as an ample divisor, Atti Accad. Ligure Sci. Lett. 38 (1982), 88–92 (English, with French summary). MR 727900
- Mauro C. Beltrametti and Andrew J. Sommese, The adjunction theory of complex projective varieties, De Gruyter Expositions in Mathematics, vol. 16, Walter de Gruyter & Co., Berlin, 1995. MR 1318687, DOI 10.1515/9783110871746
- Takao Fujita, Classification theories of polarized varieties, London Mathematical Society Lecture Note Series, vol. 155, Cambridge University Press, Cambridge, 1990. MR 1162108, DOI 10.1017/CBO9780511662638
- Takao Fujita, On polarized manifolds whose adjoint bundles are not semipositive, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 167–178. MR 946238, DOI 10.2969/aspm/01010167
- Takao Fujita, Remarks on quasi-polarized varieties, Nagoya Math. J. 115 (1989), 105–123. MR 1018086, DOI 10.1017/S0027763000001562
- Yoshiaki Fukuma, A lower bound for the sectional genus of quasi-polarized surfaces, Geom. Dedicata 64 (1997), no. 2, 229–251. MR 1436766, DOI 10.1023/A:1004939700290
- —, A lower bound for sectional genus of quasi-polarized manifolds, J. Math. Soc. Japan 49 (1997), 339–362.
- —, A lower bound for sectional genus of quasi-polarized manifolds II, preprint.
- Yoshiaki Fukuma, On sectional genus of quasi-polarized manifolds with non-negative Kodaira dimension, Math. Nachr. 180 (1996), 75–84. MR 1397669, DOI 10.1002/mana.3211800105
- Paltin Ionescu, Generalized adjunction and applications, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 3, 457–472. MR 830359, DOI 10.1017/S0305004100064409
- Yujiro Kawamata, Katsumi Matsuda, and Kenji Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 283–360. MR 946243, DOI 10.2969/aspm/01010283
- Hidetoshi Maeda, On polarized surfaces of sectional genus three, Sci. Papers College Arts Sci. Univ. Tokyo 37 (1988), no. 2, 103–112. MR 950622
- C. P. Ramanujam, Remarks on the Kodaira vanishing theorem, J. Indian Math. Soc. (N.S.) 36 (1972), 41–51. MR 330164
- Andrew John Sommese, On the adjunction theoretic structure of projective varieties, Complex analysis and algebraic geometry (Göttingen, 1985) Lecture Notes in Math., vol. 1194, Springer, Berlin, 1986, pp. 175–213. MR 855885, DOI 10.1007/BFb0077004
- Andrew John Sommese and A. Van de Ven, On the adjunction mapping, Math. Ann. 278 (1987), no. 1-4, 593–603. MR 909240, DOI 10.1007/BF01458083
Additional Information
- Yoshiaki Fukuma
- Affiliation: Department of Mathematics, Faculty of Science, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152, Japan
- Address at time of publication: Department of Mathematics, College of Education, Naruto University of Education, Takashima, Naruto-cho, Naruto-shi 772-8502, Japan
- Email: fukuma@naruto-u.ac.jp
- Received by editor(s): February 5, 1997
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 351 (1999), 363-377
- MSC (1991): Primary 14C20; Secondary 14J99
- DOI: https://doi.org/10.1090/S0002-9947-99-02235-7
- MathSciNet review: 1487615