## Tohoku Mathematical Journal 2002 December SECOND SERIES VOL. 54, NO. 4

 Tohoku Math. J. 54 (2002), 593-597

Title TORIC VARIETIES WHOSE BLOW-UP AT A POINT IS FANO

Author Laurent Bonavero

(Received December 26, 2000, revised August 20, 2001)
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.

2000 Mathematics Subject Classification. Primary 14E30; Secondary 14J45, 14M25.

Key words and phrases. Toric Fano varieties, blow-up, Mori theory.