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Let $X,Y,$ and $Z$ be schemes, $Y \hookrightarrow Z$ an immersion so that $Y$ is locally closed in $Z$, and let $\overline{Y}$ be the closure of $Y$ in $Z$. Under what conditions if any can a morphism $f:X \times Y \to Y$ extend to a morphism $X \times \overline{Y} \to \overline{Y}$? I'm looking for both conditions on the $X,Y$, and $Z$ or on $f$ itself, really anything that can be said about this type of situation. The type of thing I had in mind was a group action extending to the closure. The specific case I was looking at are if $X$ and $Y$ are both an algebraic torus $T$, when does the action $T \times T \to T$ extend to $T \times \overline{T} \to \overline{T}$ but really anything that can be said about this at any level of generality would be appreciated.

This question is inspired from my more specific question Are the closures of the tori in the decomposition of a torified variety toric varieties?

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This is not a comment on the question so much as on your intended application. In the most common definition of a toric variety, such as in Fulton's book, the variety is also assumed to normal. The examples in your previous question gave examples of torified varieties where the torus failed to act on the whole variety. However, extending the action of the torus to the whole variety, as in this question, is not enough. You'll also need the variety to be normal in order to apply results from toric geometry. – Dustin Cartwright Sep 4 '11 at 19:51
If $Y = \overline Y$. HA! – Taylor Dupuy Sep 8 '11 at 17:44

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