## Tokyo Journal of Mathematics

### On a Generalization of the Mukai Conjecture for Fano Fourfolds

Kento FUJITA

#### Abstract

Let $X$ be a complex $n$-dimensional Fano manifold. Let $s(X)$ be the sum of $l(R)-1$ for all the extremal rays $R$ of $X$, the edges of the cone $\operatorname{NE}(X)$ of curves of $X$, where $l(R)$ denotes the minimum of $(-K_X \cdot C)$ for all rational curves $C$ whose classes $[C]$ belong to $R$. We show that $s(X)\leq n$ if $n\leq 4$. And for $n\leq 4$, we completely classify the case the equality holds. This is a refinement of the Mukai conjecture on Fano fourfolds.

#### Article information

Source
Tokyo J. Math., Volume 37, Number 2 (2014), 319-333.

Dates
First available in Project Euclid: 28 January 2015

https://projecteuclid.org/euclid.tjm/1422452796

Digital Object Identifier
doi:10.3836/tjm/1422452796

Mathematical Reviews number (MathSciNet)
MR3304684

Zentralblatt MATH identifier
1305.62035

Subjects
Primary: 14J45: Fano varieties
Secondary: 14E30: Minimal model program (Mori theory, extremal rays) 14J35: $4$-folds

#### Citation

FUJITA, Kento. On a Generalization of the Mukai Conjecture for Fano Fourfolds. Tokyo J. Math. 37 (2014), no. 2, 319--333. doi:10.3836/tjm/1422452796. https://projecteuclid.org/euclid.tjm/1422452796

#### References

• M. Andreatta, E. Chierici and G. Occhetta, Generalized Mukai conjecture for special Fano varieties, Cent. Eur. J. Math. 2 (2004), no. 2, 272–293.
• M. Andreatta and G. Occhetta, Special rays in the Mori cone of a projective variety, Nagoya Math. J. 168 (2002), 127–137.
• C. Araujo, Rational curves of minimal degree and characterizations of projective spaces, Math. Ann. 335 (2006), no. 4, 937–951.
• L. Bonavero, C. Casagrande, O. Debarre and S. Druel, Surune conjecture de Mukai, Comment. Math. Helv. 78 (2003), no. 3, 616–626.
• L. Bonavero, F. Campana and J. A. Wiśniewski, Variétés projectives complexes dont l'éclatée en un point est de Fano, C. R. Math. Acad. Sci. Paris 334 (2002), no. 6, 463–468.
• C. Casagrande, The number of vertices of a Fano polytope, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 1, 121–130.
• C. Casagrande, Quasi-elementary contractions of Fano manifolds, Compos. Math. 144 (2008), no. 6, 1429–1460.
• C. Casagrande, On Fano manifolds with a birational contraction sending a divisor to a curve, Michigan Math. J. 58 (2009), no. 3, 783–805.
• K. Cho, Y. Miyaoka and N. I. Shepherd-Barron, Characterizations of projective spaces and applications to complex symplectic manifolds, Higher dimensional birational geometry (Kyoto, 1997), 1–88, Adv. Stud. Pure Math. 35, Math. Soc. Japan, Tokyo, 2002.
• R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heigelberg, 1977.
• S. Kebekus, Families of singular rational curves, J. Algebraic Geom. 11 (2002), no. 2, 245–256.
• J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Math, vol. 134, Cambridge University Press, Cambridge, 1998.
• J. Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 32, Springer-Verlag, 1996.
• S. Mori and S. Mukai, Classification of Fano $3$-folds with $B_2\geq 2$, Manuscripta Math. 36 (1981), no. 2, 147–162. Erratum: 110 (2003), no. 3, 407.
• S. Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. (2) 116 (1982), no. 1, 133–176.
• S. Mukai, Problems on characterization of the complex projective space, Birational Geometry of Algebraic Varieties, Open Problems, Proceedings of the 23rd Symposium of the Taniguchi Foundation at Katata, Japan, 1988, pp. 57–60.
• C. Novelli and G. Occhetta, Rational curves and bounds on the Picard number of Fano manifolds, Geom. Dedicata 147 (2010), 207–217.
• G. Occhetta, A characterization of products of projective spaces, Canad. Math. Bull. 49 (2006), no. 2, 270–280.
• T. Tsukioka, Fano manifolds obtained by blowing up along curves with maximal Picard number, Manuscripta Math. 132 (2010), no. 1–2, 247–255.
• T. Tsukioka, A remark on Fano $4$-folds having $(3,1)$-type extremal contractions, Math. Ann. 348 (2010), no. 3, 737–747.
• T. Tsukioka, Pseudo-index and the length of extremal rays of Fano manifolds (in Japanese), WebProceedings of Mini-Conference on Algebraic Geometry in Saitama University, March 1–2, 2010. See http://www.rimath.saitama-u.ac.jp/lab.jp/fsakai/proc2010e.html/
• T. Tsukioka, On the minimal length of extremal rays for Fano four-folds, Math. Z. 271 (2012), no. 1–2, 555–564.
• J. A. Wiśniewski, On a conjecture of Mukai, Manuscripta Math. 68 (1990), no. 2, 135–141.
• J. A. Wiśniewski, On contractions of extremal rays of Fano manifolds, J. Reine Angew. Math. 417 (1991), 141–157.