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Given a torus $T$ is there way to classify all the toric varieties it gives rise to? That is, classify all toric varieties $X$ whose torus is isomorphic to $T$. Is there a way to construct these toric varieties (i.e. give equations for them)?

Edit

Remark: as has been explained in the comments and answers, this question is I'll posed. But it seems to have generated good discussion so I'm leaving the phrasing as is.

2 added 173 characters in body

Given a torus $T$ is there way to classify all the toric varieties it gives rise to? That is, classify all toric varieties $X$ whose torus is isomorphic to $T$. Is there a way to construct these toric varieties (i.e. give equations for them)?

Edit: as has been explained in the comments and answers, this question is I'll posed. But it seems to have generated good discussion so I'm leaving the phrasing as is.

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# Counting/constructing Toric Varieties

Given a torus $T$ is there way to classify all the toric varieties it gives rise to? That is, classify all toric varieties $X$ whose torus is isomorphic to $T$. Is there a way to construct these toric varieties (i.e. give equations for them)?