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In this link, 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 someone please illustrate this 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. |
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Merely observe that a toric variety is the union of torus orbits $(\mathbb C^*)^r$ for various dimensions $r$, and that the Euler characteristic of $(\mathbb C^*)^r$ is zero if $r>0$ and $1$ if $r=0$. Vertices of a polytope correspond to 0-dimensional orbits, $r$-dimensional faces -- to $r$-dimensional orbits. $\mathbb P^2$ corresponds to a triangle, $\mathbb P^1\times\mathbb P^1$ to a square. It is not very hard to count their vertices. |
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