On the pluricanonical map of threefolds of general type
HTML articles powered by AMS MathViewer
- by Dong-Kwan Shin PDF
- Proc. Amer. Math. Soc. 124 (1996), 3641-3646 Request permission
Abstract:
Let $X$ be a smooth minimal threefold of general type and let $n$ be an integer $>1$. Assume that the image of the pluricanonical map $\Phi _{n}$ of $X$ is a curve. Then a simple computation shows that $n$ is necessarily $2$ or $3$. When $n=2$ with a numerical condition or when $n=3$, we obtain two inequalities $\chi (\mathcal {O}_{X})\leq \text {min}\{-1,2-2q_{1}\}$ and $q_{1}\leq \dfrac {3}{14}{K_{X}}^{3}+1$, where $q_{1}$ is the irregularity of $X$ and $\chi (\mathcal {O}_{X})$ is the Euler characteristic of $X$.References
- E. Bombieri, Canonical models of surfaces of general type, Inst. Hautes Études Sci. Publ. Math. 42 (1973), 171–219. MR 318163, DOI 10.1007/BF02685880
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
- Yujiro Kawamata, Hodge theory and Kodaira dimension, Algebraic varieties and analytic varieties (Tokyo, 1981) Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp. 317–327. MR 715655, DOI 10.2969/aspm/00110317
- 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
- János Kollár, Higher direct images of dualizing sheaves. I, Ann. of Math. (2) 123 (1986), no. 1, 11–42. MR 825838, DOI 10.2307/1971351
- Radu Bǎdescu, On a problem of Goursat, Gaz. Mat. 44 (1939), 571–577. MR 0000087
- Yoichi Miyaoka, The Chern classes and Kodaira dimension of a minimal variety, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 449–476. MR 946247, DOI 10.2969/aspm/01010449
- Kenji Ueno, Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Mathematics, Vol. 439, Springer-Verlag, Berlin-New York, 1975. Notes written in collaboration with P. Cherenack. MR 0506253, DOI 10.1007/BFb0070570
Additional Information
- Dong-Kwan Shin
- Affiliation: Department of Mathematics, Konkuk University, Seoul, 143–701, Korea
- Email: shindk@cs.sejong.ac.kr
- Received by editor(s): June 12, 1995
- Additional Notes: This paper is supported by KOSEF and Dae-Yang Foundation
- Communicated by: Eric M. Friedlander
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 3641-3646
- MSC (1991): Primary 14E05, 14J30
- DOI: https://doi.org/10.1090/S0002-9939-96-03865-8
- MathSciNet review: 1389536