# Why the blowup of a toric variety corresponding to a subdivison of fan?

I heard the statement "the blowup of a toric variety corresponding to a subdivison of fan" many times, but could not find reference in the literature. What is the precised statement (blowup a point? or certain subscheme?)? And how does blowup related to the fan?

I think one can just consider an affine toric variety defined by a cone $\sigma$.

• Cox, Little and Schenck talk about this in their Toric Varieties book. See for instance Chapter 3. Aug 7 '13 at 4:18
• This question appears to be off-topic because it is about Answered in the comments Aug 7 '13 at 11:10

If $\Delta$ is a fan in a lattice $N$, and $\sigma \in \Delta$ is a cone, the star of $\sigma$ $-$ call it $\Delta'$ $-$ is a refinement of $\Delta$. Then the morphism $X(\Delta') \to X(\Delta)$ of toric varieties induced by identity map of $N$ exhibits $X(\Delta')$ as the blowup of $X(\Delta)$ at the distinguished point $x_\sigma$ (fixed point of the torus action).