## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### Fano manifolds which are not slope stable along curves

Kento Fujita

#### Abstract

We show that a Fano manifold $(X,-K_{X})$ is \textit{not} slope stable with respect to a smooth curve $Z$ if and only if $(X,Z)$ is isomorphic to one of (projective space, line), (product of projective line and projective space, fiber of second projection) or (blow up of projective space along linear subspace of codimension two, nontrivial fiber of blow up).

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 87, Number 10 (2011), 199-202.

Dates
First available in Project Euclid: 1 December 2011

https://projecteuclid.org/euclid.pja/1322748850

Digital Object Identifier
doi:10.3792/pjaa.87.199

Mathematical Reviews number (MathSciNet)
MR2863414

Zentralblatt MATH identifier
1236.14040

#### Citation

Fujita, Kento. Fano manifolds which are not slope stable along curves. Proc. Japan Acad. Ser. A Math. Sci. 87 (2011), no. 10, 199--202. doi:10.3792/pjaa.87.199. https://projecteuclid.org/euclid.pja/1322748850

#### 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.
• J.-M. Hwang, H. Kim, Y. Lee and J. Park, Slopes of smooth curves on Fano manifolds, Bull. London Math. Soc., Published online.
• J. Kollár and S. Mori, Birational geometry of algebraic varieties, translated from the 1998 Japanese original, Cambridge Tracts in Mathematics, 134, Cambridge Univ. Press, Cambridge, 1998.
• J. Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 32, Springer, Berlin, 1996.
• S. Mori and S. Mukai, Classification of Fano 3-folds with $B_{\text{2}}\geq \text{2}$, Manuscr. Math. 36 (1981), no. 2, 147–162. Erratum: 110 (2003), no. 3, 407.
• S. Mori and S. Mukai, On Fano 3-folds with $B_{2}\geq 2$, in Algebraic varieties and analytic varieties (Tokyo, 1981), 101–129, Adv. Stud. Pure Math., 1, North-Holland, Amsterdam, 1983.
• J. Ross and R. Thomas, A study of the Hilbert-Mumford criterion for the stability of projective varieties, J. Algebraic Geom. 16 (2007), no. 2, 201–255.
• T. Tsukioka, On the minimal length of extremal rays for Fano four-folds, Math. Z., Published online.
• J. A. Wiśniewski, On contractions of extremal rays of Fano manifolds, J. Reine Angew. Math. 417 (1991), 141–157.