Are Grassmannians $G(k,n)$ toric varieties for all possible $k,n$? If they are toric varieties, are there any descriptions for the fans?
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12$\begingroup$ Look at page 5 of math.mit.edu/~mckernan/Teaching/10-11/Spring/18.726/l_4.pdf for a negative answer to your first question. BTW: This was proof by Google. It is best to Google first and ask MO questions later! $\endgroup$– Jon BannonSep 1, 2013 at 20:36
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5$\begingroup$ The Grassmannian $G(k,n)$ is toric if and only if $k = 1$ or $k = n - 1$, in which case it is isomorphic to $\mathbb{P}^{n-1}$. Instead of having a dense orbit for a torus in general, it always has a dense orbit for a Borel subgroup of $SL(n)$ (e.g. the subgroup of upper triangular matrices of determinant 1). $\endgroup$– Michael JoyceSep 1, 2013 at 20:55
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$\begingroup$ I did not know this. $\endgroup$– john mangualSep 1, 2013 at 20:56
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$\begingroup$ @Jon Bannon Thank you! But why "the toric variety in $\mathbb{P}^5$ must be defined by a binomial" as commented in the article you cited. $\endgroup$– Li YutongSep 1, 2013 at 21:00
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$\begingroup$ @MichaelJoyce Thank you for your answer! $\endgroup$– Li YutongSep 1, 2013 at 21:03
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