Hi all

in this the following link (http://www.math.leidenuniv.nl/scripties/Trevisan.pdf), Corollary 3.2.2, page 59 the author claims that : The Euler characteristic of the toric variety $X_K$ associated to a convex polytope $K$ is the number of vertices of $K$.

I want to see how it works, could some one please illustrate for me by using this method to compute the Euler characteristic of $\mathbb{P}^{2}$ and $\mathbb{P}^{1}\times \mathbb{P}^{1}$. thanks so much