# 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 ${\displaystyle \mathbb {P} ^{2}}$ 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 ${\displaystyle X}$ is an affine toric variety over ${\displaystyle \mathbb {C} }$, of dimension ${\displaystyle n}$. By construction, ${\displaystyle X}$ has an open orbit isomorphic to ${\displaystyle (\mathbb {C} ^{\ast })^{n}.}$ Further, the ${\displaystyle \mathbb {Z} }$-module ${\displaystyle \mathrm {Hom} (\mathbb {C} ^{\ast },(\mathbb {C} ^{\ast })^{n})}$ of homomorphisms of algebraic groups is free of rank ${\displaystyle n}$. We say that an element ${\displaystyle \psi \in \mathrm {Hom} (\mathbb {C} ^{\ast },(\mathbb {C} ^{\ast })^{n})}$converges in ${\displaystyle X}$ if there is a map ${\displaystyle {\bar {\psi }}\colon \mathbb {C} \to X\supseteq (\mathbb {C} ^{\ast })^{n}}$ such that

${\displaystyle {\bar {\psi }}\mid _{\mathbb {C} ^{\ast }}=\psi .}$
Define the polyhedral cone associated to ${\displaystyle X}$ to be the cone
${\displaystyle \sigma \subseteq \mathrm {Hom} (\mathbb {C} ^{\ast },(\mathbb {C} ^{\ast })^{n})\otimes _{\mathbb {Z} }\mathbb {Q} }$
generated by the convergent morphisms. If ${\displaystyle X}$ is not necessarily affine, we can cover ${\displaystyle X}$ 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 ${\displaystyle \mathbb {C} ^{n}}$ and ${\displaystyle \mathbb {P} ^{n}}$ with the usual torus actions.

Now let us consider the variety

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