# Toric blow-up

In this section, we will consider the blow-up of toric varieties, and in particular how the combinatorial description (polytope or fan) changes under this operation. We will particularly consider the example of $\mathbb {P} ^{2}$ with the usual action of the torus, looking both at the symplectic and the algebro-geometric side of the theory. Let us first recall briefly the construction of the fan associated to a toric variety. Suppose $X$ is an affine toric variety over $\mathbb {C}$ , of dimension $n$ . By construction, $X$ has an open orbit isomorphic to $(\mathbb {C} ^{\ast })^{n}.$ Further, the $\mathbb {Z}$ -module $\mathrm {Hom} (\mathbb {C} ^{\ast },(\mathbb {C} ^{\ast })^{n})$ of homomorphisms of algebraic groups is free of rank $n$ . We say that an element $\psi \in \mathrm {Hom} (\mathbb {C} ^{\ast },(\mathbb {C} ^{\ast })^{n})$ converges in $X$ if there is a map ${\bar {\psi }}\colon \mathbb {C} \to X\supseteq (\mathbb {C} ^{\ast })^{n}$ such that

${\bar {\psi }}\mid _{\mathbb {C} ^{\ast }}=\psi .$ Define the polyhedral cone associated to $X$ to be the cone
$\sigma \subseteq \mathrm {Hom} (\mathbb {C} ^{\ast },(\mathbb {C} ^{\ast })^{n})\otimes _{\mathbb {Z} }\mathbb {Q}$ generated by the convergent morphisms. If $X$ is not necessarily affine, we can cover $X$ with finitely many open affine toric subvarieties, and we can patch the polyhedral cones together to obtain a fan . As it turns out, this combinatorial object determines the toric variety up to isomorphism.

Exercise 8.2

Find the polyhedral cone associated to $\mathbb {C} ^{n}$ and $\mathbb {P} ^{n}$ with the usual torus actions.

Now let us consider the variety

$\mathrm {Bl} _{0}\,\mathbb {C} ^{2}:=\{((x,y),[a:b])\in \mathbb {C} ^{2}\times _{\mathbb {C} }\mathbb {P} ^{1}\colon ax-by=0\}$ with the torus action
$(t_{1},t_{2})\cdot ((x,y),[a:b]):=((t_{1}x,t_{2}y),[t_{1}a:t_{2}b]).$ This variety can be covered by open affine toric subvarieties given by $a\not =0$ and $b\not =0$ . We know that any homomorphism of algebraic groups
$\mathbb {C} ^{\ast }\to (\mathbb {C} ^{\ast })^{2}$ is given by
$t\mapsto (t^{k_{1}},t^{k_{2}})$ with $k_{j}\in \mathbb {Z}$ for $j=1,2$ . Therefore, in order for such a morphism to converge, we must have $k_{i}\geq 0$ for $i=1,2$ . Furthermore, we have
$[t^{k_{1}}:t^{k_{2}}]=[1:t^{k_{2}-k_{1}}],$ so (as "$[1:\infty ]=[0:1]$ "), we also obtain the condition that $k_{2}\geq k_{1}$ . Putting things together, we find that the fan associated to $\mathrm {Bl} _{0}\,\mathbb {C} ^{2}$ consists of the cones generated by $(1,0),(1,1)$ and $(1,1),(0,1)$ . Using a very similar argument, we can show that the fan associated to $\mathrm {Bl} _{[0:1:1]}\,\mathbb {P} ^{2}$ consists of the cones generated by $(1,0),(1,1)$ , $(1,1),(0,1)$ , $(0,1),(-1,-1)$ and $(-1,-1),(1,0)$ , so it arises from the fan of $\mathbb {P} ^{2}$ by adding an extra ray. This corresponds to the fact that the polytope of a blow-up (in a fixed point) corresponds to chopping off the vertex coming from this fixed point.