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In dimension 1, the only smooth projective toric Fano variety is $\mathbb{P}^1$. In dimension $2$, there are 5: $\mathbb{P}^1\times \mathbb{P}^1$, and then successive blow-ups of $\mathbb{P}^2$ at up to 3 torus-invariant points. In dimension 3 there are 18, in dimension 4, 124. I am wondering whether there is a formula in terms of the dimension, $n$, which gives a bound on the number of smooth projective toric Fano varieties of dimension $n$.

I believe that adding in the Fano condition is necessary to have a finite number (i.e., if we just asked for smooth and projective, then I believe I can get infinitely many non-isomorphic in dimension $\geq 2$).

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    $\begingroup$ The number of smooth projective toric Fano varieties is known up to dimension 9 by work of Mikkel Øbro, cf. sequence A140296 in OEIS. No formula or good upper bound seems to be known. $\endgroup$
    – Kasper Andersen
    Commented Sep 3 at 21:49
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    $\begingroup$ There are known upper bounds on the number of vertices of smooth Fano polytopes, so if you combine that with the known bounds on number of combinatorial types of polytopes with given dimension and number of vertices you get a (very large) upper bound. Is that the sort of thing you are looking for? $\endgroup$
    – Gjergji Zaimi
    Commented Sep 3 at 21:54
  • $\begingroup$ @Gjergji: A very large upper bound will do! I see from Casagrande's "Number of Vertices of a Fano polytope" that $|V(P)| \leq 3n$, and from Alon's "Number of Polytopes, Configurations, and Real Matroids" that the number of distinct labeled d polytopes on n vertices is at most $(n/d)^{d^2*n(1+o(1))}$ as $n/d \rightarrow \infty$, so I can almost combine these to $3^{d^2*(3d)(1+o(1))}$, except that $n/d$ is fixed at 3 in the bound, rather than $\rightarrow\infty$. I will keep looking for better bounds though, unless you know of any? $\endgroup$
    – locally trivial
    Commented Sep 3 at 23:25
  • $\begingroup$ I see Goodman-Pollack give $n^{d(d+1)n}$ for simplicial polytopes in dimension $d$ with $n$ vertices. I should be able to do something with this... $\endgroup$
    – locally trivial
    Commented Sep 3 at 23:37
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    $\begingroup$ Agh! I was actually hoping to be able to take a limit of the partition number $p(d) \sim \frac{1}{4d\sqrt{3}} \exp{\pi \sqrt{(2d)/3}}$ divided by the estimate above, and get something non-zero. But $(3d)^{3d^3+3d^2}$ will grow much faster than $p(d)$, so I will look for better bounds. But indeed: you have answered my question, thank you. Note: This limit being $0$ suggests, but does not prove, that products of projective spaces make up $0\%$ of all smooth projective toric Fano varieties in the limit, which we might expect. $\endgroup$
    – locally trivial
    Commented Sep 4 at 0:06

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