## Tohoku Mathematical Journal

### Toric varieties whose blow-up at a point is Fano

Laurent Bonavero

#### Abstract

We classify smooth toric Fano varieties of dimension $n\geq 3$ containing a toric divisor isomorphic to the $(n-1)$-dimensional projective space. As a consequence of this classification, we show that any smooth complete toric variety $X$ of dimension $n\geq 3$ with a fixed point $x\in X$ such that the blow-up $B_x(X)$ of $X$ at $x$ is Fano is isomorphic either to the $n$-dimensional projective space or to the blow-up of the $n$-dimensional projective space along an invariant linear codimension two subspace. As expected, such results are proved using toric Mori theory due to Reid.

#### Article information

Source
Tohoku Math. J. (2), Volume 54, Number 4 (2002), 593-597.

Dates
First available in Project Euclid: 11 April 2005

https://projecteuclid.org/euclid.tmj/1113247651

Digital Object Identifier
doi:10.2748/tmj/1113247651

Mathematical Reviews number (MathSciNet)
MR1936270

Zentralblatt MATH identifier
1021.14014

#### Citation

Bonavero, Laurent. Toric varieties whose blow-up at a point is Fano. Tohoku Math. J. (2) 54 (2002), no. 4, 593--597. doi:10.2748/tmj/1113247651. https://projecteuclid.org/euclid.tmj/1113247651

#### References

• V.V. Batyrev, Toroidal Fano 3-folds, Math. USSR, Izv. 19 (1982), 13–25.
• V.V. Batyrev, On the classification of toric Fano $4$-folds, Algebraic geometry, 9. J. Math. Sci. 94 (1999), 1021–1050.
• C. Casagrande, On the birationnal geometry of toric Fano $4$-folds, C. R. Acad. Sci. Paris Sr. I Math. 332 (2001), no. 12, 1093–1098.
• W. Fulton, Introduction to toric varieties, Ann. of Math. Stud. 131, Princeton University Press, Princeton, NJ, 1993.
• J. Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihre Grenzgebiete (3) 32, 2nd edition, Springer, Berlin, 1999.
• T. Oda, Convex bodies and algebraic geometry, An introduction to the theory of toric varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 15, Springer, Berlin, 1988.
• M. Reid, Decomposition of toric morphisms, Arithmetic and geometry, Vol. II. 395–418, Prog. Math. 36, Birkhäuser Boston, Boston, MA, 1983.
• H. Sato, Toward the classification of higher-dimensional toric Fano varieties, Tohoku Math. J. 52 (2000), 383–413.
• K. Watanabe and M. Watanabe, The classification of Fano $3$-folds with torus embeddings, Tokyo J. Math. 5 (1982), 37–48.