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Part of the definition of an affine toric variety is that the action of the torus sitting as an open dense subset of the variety extends algebraically to the whole variety. Is there an easy example of a variety which contains an open subset isomorphic to a torus for which the action cannot be extended algebraically?

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Blow up two or more distinct points on a line in $\mathbb{A}^2$. – MP Jan 26 2012 at 0:04
MP: I think he wants an affine example. – J.C. Ottem Jan 26 2012 at 0:10
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 Ok, then glue together two distinct points in $\mathbb{A}^2$. – MP Jan 26 2012 at 0:21
What do you mean by glue together points? Will this still be separated and irreducible? – HNuer Jan 26 2012 at 17:00
You just need to define the local ring at the glued spot. If you choose to specify that the intersection is transverse, then you will get a separated irreducible scheme. – S. Carnahan♦ Jan 27 2012 at 9:41

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