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I'm looking for a reference on the following question:

Given a fixed toric variety $V/k$, how to describe the moduli space of all toric structures on $V$?

In addition to the general question, I would also be interested to see a description for the special cases where $V\cong \mathbb{P}^n$ or where the torus embedding $(k^{\times})^n \hookrightarrow V$ has been fixed, and the action is allow to vary.

Thank you.

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    $\begingroup$ Some comments: 1) a toric str is “the same” as a divisor $D\sim -K_X$ st $\Omega_X(\log D)$ is trivial, so your moduli sp should “live” in $|-K_X|$, 2) automorphisms were computed by Cox, 3) if you fix the torus, the toric str is unique if it exists $\endgroup$
    – Piotr Achinger
    Commented Apr 22, 2019 at 20:38
  • $\begingroup$ @PiotrAchinger Thanks for your remarks. Can you please refer me to a further explanation of your 1)? $\endgroup$
    – Somatic Custard
    Commented Apr 23, 2019 at 20:31
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    $\begingroup$ Demazure has a paper about the automorphism group of toric varieties, which can be nonreductive, e.g. $\mathbb P^2$ blown up at a point. Then I guess your question is about maximal tori of that automorphism group? They're still all conjugate as I recall, i.e., the resulting toric varieties are equivariantly isomorphic. Things change a great deal if you consider different toric variety structures on the same real manifold, e.g., all the even Hirzebruch surfaces are toric and complexly different but really diffeomorphic. $\endgroup$
    – Allen Knutson
    Commented May 18, 2019 at 23:13

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