×

Classification of logarithmic Fano threefolds. (English) Zbl 0658.14019

Throughout this paper varieties are defined over a fixed algebraically closed field of characteristic zero. A logarithmic Fano threefold is defined to be a pair (V,D) of a smooth projective threefold V and a reduced divisor D with normal crossings on V, satisfying the following condition: \(-K_ V-D\quad is\quad ample.\)
The purpose of this paper is to classify logarithmic Fano threefolds (V,D) with non-zero boundaries D. Fundamental tools are Norimatsu’s vanishing theorem, Tsunoda’s logarithmic cone theorem, Mori’s theory of extremal rational curves on a threefold and some ampleness criteria for the logarithmic anti-canonical divisor, which will be explained in section 1.
In section 2 we study some general properties of logarithmic Fano varieties (V,D) of arbitrary dimension.
In section 4 we prove the existence of an extremal rational curve \(\ell\) with \((D\cdot \ell)>0\) as a key lemma. Moreover all the types of \(\ell\) are F, \(E_ 2, D_ 3, D_ 2\) or \(C_ 2\). - Logarithmic Fano threefolds (V,D) with \(D\neq 0\) are classified into five types.
We give in section 5 the configuration of boundaries.
From sections 6 to 9 we classify logarithmic Fano threefolds according to the types of extremal rational curves.

MSC:

14J30 \(3\)-folds
14J10 Families, moduli, classification: algebraic theory
PDF BibTeX XML Cite
Full Text: Numdam EuDML

References:

[1] I.V. Dëmin : Fano 3-folds representable in the form of line bundles . Math. USSR Izvestija 17 (1981) 219-226. · Zbl 0536.14025
[2] I.V. Dëmin : Addemdum to the paper ”Fano 3-folds representable in the form of line bundles” . Math. USSR Izvestija 20 (1983) 625-626. · Zbl 0536.14026
[3] T. Fujita : On the structure of polarized varieties with \Delta -genera zero . J. Fac. Sci. Univ. Tokyo 22 (1975) 103-115. · Zbl 0333.14004
[4] T. Fujita : On topological characterizations of complex projective spaces and affine linear spaces . Proc. Japan Acad. 56 (1980) 231-234. · Zbl 0453.14008
[5] R. Hartshorne : Algebraic Geometry . GTM 52, Springer (1977). · Zbl 0367.14001
[6] S. Iitaka : Algebraic Geometry: An Introduction to Birational Geometry of Algebraic Varieties . GTM 76, Springer (1981). · Zbl 0491.14006
[7] S. Iitaka : Birational Geometry for Open Algebraic Varieties . Montreal UP 76 (1981). · Zbl 0491.14005
[8] V.A. Iskovskih : Anticanonical models of three dimensional algebraic varieties . J. Soviet Math. 13 (1980) 815-868. · Zbl 0428.14016
[9] V.A. Iskovskih : Treedimensional Fano varities. I, II . Math. USSR Izvestija 11 (1977) 485-527, · Zbl 0382.14013
[10] ibid., 12 (1978) 469-509.
[11] K. Kodaira : On stability of compact submanifolds of complex manifolds . Amer. J. Math. 85 (1963) 79-94. · Zbl 0173.33101
[12] S.L. Kleiman : Toward a numberical theory of ampleness . Ann. of Math. 84 (1966) 293-344. · Zbl 0146.17001
[13] H. Maeda : On certain ampleness criteria for divisors on threefolds . Preprint (1983).
[14] Y.I. Manin : Cubic Forms: Algebra, Geometry, Arithmetic . North-Holland (1974). · Zbl 0277.14014
[15] M. Miyanishi : Algebraic methods in the theory of algebraic threefolds . In: Algebraic Varieties and Analytic Varities, Advanced Studies in Pure Mathematics . 1. North-Holland Kinokuniya (1983). · Zbl 0537.14027
[16] S. Mori : Threefolds whose canonical bundles are not numerically effectie . Ann. of Math. 116 (1982) 133-176. · Zbl 0557.14021
[17] S. Mori and S. Mukai : Classifications of 3-folds with B2 \succcurleq 2 . Manuscripta Math. 36 (1981) 147-162. · Zbl 0478.14033
[18] S. Mori and S. Mukai : On Fano 3-folds with B2 \succcurleq 2. In: Algebraic Varieties and Analytic Varieties , Advanced Studies in Pure Math. 1. North-Holland Kinokuniya (1983). · Zbl 0537.14026
[19] J.P. Murre : Classification of Fano threefolds according to Fano and Iskovskih . In: Algebraic Threefolds: Lect. Notes in Math. 947. Springer (1981). · Zbl 0492.14025
[20] Y. Norimatsu : Kodaira vanishing theorem and Chern classes for \partial -manifolds . Proc. J. Acad. 54 (1978) 107-108. · Zbl 0433.32013
[21] N. Nygaard : On the fundamental group of uni-rational 3-fold . Invent. math. 44 (1978) 75-86. · Zbl 0427.14014
[22] M. Reid : Lines on Fano 3-folds according to Shokurov . Mittag-Leffler Report No. 11 (1980).
[23] S. Tsunoda : Open surfaces of logarithmic Kodaira dimension - \infty . Talk at Seminar on Analytic Manifolds , Univ. of Tokyo (1981).
[24] S. Tsunoda and M. Miyanishi : The structure of open algebraic surfaces. II . In : Classification of Algebraic and Analytic Manifolds. Progress in Mathematics 39. Birkäuser (1983). · Zbl 0605.14035
[25] H. Umemura and S. Mukai : Minimal rational threefolds . In: Algebraic Geometry . Lect. Notes in Math. 1016. Springer (1983) · Zbl 0526.14006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.