Toric varieties whose blow-up at a point is Fano

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.