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 $\sigma$.

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

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

  • $\begingroup$ Thank you!I have noticed the related discussion in the book you mentioned above. $\endgroup$
    – Li Yutong
    Aug 7 '13 at 13:05

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