Corresponding toric variety for n-simplex Let $P $ be a Delzant polytope and $X_P $ be a corresponding Toric variety. I want to see if $P=\sum $ be a n-simplex then $X_P=\mathbb P^n$
 A: (I am answering from algebro-geometric perspective; it would be nice to see answers from more symplectically-minded people)
You must have meant the polytope $\Delta_n:=\{ (x_1, \ldots, x_n) \mid
\sum x_i \leq 1, x_i \geq 0 \}$, because the toric variety
corresponding to a lattice polytope in the $n$-dimensional character
lattice is projective if and only if the polytope is $n$-dimensional.
To contstruct a toric variety associated to a polytope in the lattice
$M_R$ dual to the character lattice one consructs a normal fan in
$N_R$. The toric variety is then glued from semigroup rings of
semigroups of points lying in dual cones. So in order to work out the
generators of these semigroups it suffices look at all the vertices
and write down the vectors that span the angle (translated to the
origin). In case of $\Delta_n$ the affine toric varieties are
isomorphic to $\mathbb{A}^n$ with coordinates
$$
(x_1, \ldots, x_n), (x_1^{-1}, x^{-1}x_2, \ldots, x^{-1}x_n), \ldots,
(x_n^{-1}x_1, x_n^{-1}x_2, \ldots, x_n^{-1})
$$
They are glued along $\mathbb{A}^n \setminus \mathbb{A}^{n-1}$-s to form
$\mathbb{P}^n$.
