Toric Fano manifolds with Picard number 1

Question

As far as I know, toric Fano manifolds are classified only up to dimension 4. In dimension one the projective line is the only example. In dimension two we have five examples: $\mathbb P ^2$, $\mathbb P^1 \times \mathbb P^1$ and $\mathbb P^2$ blown-up at 1,2 or 3 points in general position.

Toric Fano 3-folds were classified by K. Watanabe and M. Watanabe. Their result asserts that there are 18 types of such a manifold and that all of them have Picard number $\rho \leq 5$, the projective space $\mathbb P^3$ being the only one with $\rho =1$.

In four dimensions, V. Batyrev showed that there are 123 types of toric Fano 4-folds. In this case $\rho \leq 8$ and again the projective space is the only one with $\rho =1$

Classification in higher dimensions seems to be a rather diffcult problem, but the question whether $\mathbb P^n$ is the only toric Fano manifold with $\rho =1$ may be easier to answer. Is there any result in this direction?

Answer 1

I don't think the Fano condition plays an essential role here; Any smooth toric variety with Picard number 1 is a projective space. This can be verified using usual toric geometry machinery: The condition for a toric variety $X$ to be of dimension $n$ and Picard number 1 means that the fan $\Delta$ is generated by $n+1$ rays $v_0,\ldots,v_n$, where the $v_i$ are primitive in the lattice $N$. If $X$ is smooth, any subset of $n$ vectors from $\{v_0,\ldots,v_n\}$ span the lattice $N\simeq \mathbb Z^n$, and one recovers the standard fan of $\mathbb P^n$.

If $X$ is allowed to have $\mathbb Q$-factorial singularities, then the set of such Fano varieties were shown to be bounded by Borisov and Borisov.

Answer 2

The only toric varieties with $\rho = 1$ are weighted projective space. Among them only the usual projective space $P^n$ is smooth.

Answer 3

Fake weighted projective spaces exist . They are toric varieties whith Picard number one, but they are not necessarily weighted projective spaces . Cf. W.Buczynska (arXiv : 0805.1211), A.M. Kasprzyk ("Bounds on FWPS").