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 $\sigma$.



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


This is Prop. 3.3.15 in Cox, Little, Schenck, and it's probably discussed in Fulton's book as well. 


