Corresponding toric variety for n-simplex

Question

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 $X_P=\mathbb P^n$

Answer 1

(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 $\mathbb{P}^n$.