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 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 is an affine toric variety over , of dimension . By construction, has an open orbit isomorphic to Further, the -module of homomorphisms of algebraic groups is free of rank . We say that an element converges in if there is a map such that

Define the polyhedral cone associated to to be the cone
generated by the convergent morphisms. If is not necessarily affine, we can cover 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 and with the usual torus actions.


Now let us consider the variety

with the torus action
This variety can be covered by open affine toric subvarieties given by and . We know that any homomorphism of algebraic groups
is given by
with for . Therefore, in order for such a morphism to converge, we must have for . Furthermore, we have
so (as ""), we also obtain the condition that . Putting things together, we find that the fan associated to consists of the cones generated by and . Using a very similar argument, we can show that the fan associated to consists of the cones generated by , , and , so it arises from the fan of 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.