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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)?

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.

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Yes. Any book on toric varieties will explain what a fan is, and using that combinatorial datum you can construct all of them. –  Mariano Suárez-Alvarez Aug 5 '10 at 22:28
    
As for isomorphism classes: if you want $T$-equivariant isomorphisms, then the toric varieties are isomorphic iff the have the same fan. –  Mariano Suárez-Alvarez Aug 5 '10 at 22:35
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"To construct" and "to give equations" are two different questions. In particular, there exist complete toric varieties that aren't projective, so while it's possible to construct them as abstract algebraic varieties or schemes, it doesn't make sense to talk about their equations in $\mathbb{P}^n.$ –  Victor Protsak Aug 6 '10 at 5:02
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It depends on what you mean by classification. If you mean to concretely classify them it is no doubt hopeless. See for instance the extremely involved attempt (yet far from complete) of classifying all three-dimensional smooth and proper toric varieties in Oda: Torus embeddings and classifications. –  Torsten Ekedahl Aug 6 '10 at 6:48
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up vote 8 down vote accepted

As far as my understanding goes the answer is no, and I will try to explain why and clarify the list of comments (I have little reputation so I cannot comment there) and give you a partial answer. I hope I do not patronise you, since you may now already part of it.

First of all, as Torsten said, it depends what you understand for classification. In this context a torus $T$ of dimension $r$ is always an algebraic variety isomorphic to $(\mathbb{C}^*)^r$ as a group. A complex algebraic variety $X$ of finite type is toric if there exists an embedding $\iota: (\mathbb{C}^\ast)^r \hookrightarrow X$, such that the image of $\iota$ is an open set whose Zariski closure is $X$ itself and the usual multiplication in $T=\iota((\mathbb{C}^\ast)^r)$ extends to $X$ (i.e. $T$ acts on $X$).

Think about all toric varieties. It is hard to find a complete classification, i.e. being able to give the coordinates ring for each affine patch and the morphisms among them for all toric varieties.

However, when the toric varieties we consider are normal there is a structure called the fan $\Sigma$ made out of cones. All cones live in $N_\mathbb{R}\cong N\otimes \mathbb{R}$ where $N\cong \mathbb{Z}$ is a lattice. A cone is generated by several vectors of the lattices (like a high school cone, really) and a fan is a union of cones which mainly have to satisfy that they do not overlap unless the overlap is a face of the cone (another cone of smaller dimension). There is a concept of morphism of fans and hence we can speak of fans 'up to isomorphism' (elements of $\mathbf{SL}(n,\mathbb{Z})$). Given a lattice N, there is an associated torus $T_N=N\otimes (\mathbb{C}^*)$, isomorphic to the standard torus.

Then we have a 1:1 correspondence between separated normal toric varieties $X$ (which contain the torus $T_N$ as a subset) up to isomorphism and fans in $N_\mathbb{R}$ up to isomorphism. There are algorithms to compute the fan from the variety and the variety from the fan and they are not difficult at all. You can easily learn them in chapter seven of the Mirror Symmetry book, available for free. Given any toric variety (even non-normal ones) we can compute its fan, but computing back the variety of this fan many not give us the original variety unless the original is normal. You can check this easily by computing the fan of a $\mathbf{V}(x^2-y^3)$ (torus embedding $(t^3,t^2)$) which is the same as $\mathbb{C}^1$ but obviously they are not isomorphic (the former has a singularity at (0,0)). In fact, since there are only two non-isomorphic fans of dimension 1 (the one generated by $1\in \mathbb{Z}$ and the one generated by 1 and -1) we see that there are only three normal toric varieties of dimension 1, the projective line and the affine line, and the standard torus.

The proof of this statement is not easy and to be honest I have never seen it written down complete (and I would appreciate any reference if someone saw it) but I know more or less the reason, as it is explained in the book about to be published by Cox, Little and Schenck (partly available) This theorem is part of my first year report which is due by the end of September, so if you want me to send you a copy when it is finished send me an e-mail.

So, yes, in the case of normal varieties there is some 'classification' via combinatorics, but in the case of non-normal I doubt there is (I never worked with them anyways).

Become a toric fan!.

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Welcome to MO! One should probably add that older books on toric varieties, such as Fulton, define a toric variety to be normal. In this case, Toric Varieties (up to $T$-equivariant isomorphism) are in bijection with fans. –  David Speyer Aug 13 '10 at 12:27
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Shouldn't $\mathbb{C}^*$ also appear in your list of normal toric varieties of dimension one? :) –  damiano Aug 13 '10 at 17:45
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Hum, I am not sure. I think according to the definition you have to take the Zariski Closure, which would make it just $\mathbb{C}^1$, right? You make me doubt... –  Jesus Martinez Garcia Aug 15 '10 at 15:02
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I think $T=\mathbb{C}^*$ should be on the list. $T$ is closed in the zariski top. as a sub var. of itself. Plus I think it would be weird if tori weren't toric varieties. –  solbap Aug 15 '10 at 16:50
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OK, fixed, the torus has to be a proper subset (I checked, very important for the proof) and the variety has to be separated (important too). –  Jesus Martinez Garcia Aug 16 '10 at 16:08
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This question seems based on a confusion of two different possible meanings of a "torus" in algebraic geometry. A torus could mean (but among algebraic geometers, usually doesn't) an algebraic group which is isomorphic as a real Lie group to a Cartesian product of circles. That kind of torus has a moduli space of complex algebraic structures, and in higher dimensions a larger moduli space of complex analytic structures. Or a torus could mean a Cartesian power of the non-zero complex numbers $\mathbb{C}^*$. In a natural sense, there is only one of them in each complex dimension. A toric variety is a compactification of a torus in this second sense. Thus, the $n$-dimensional torus $(\mathbb{C}^*)^n$ gives rise to all $n$-dimensional toric varieties.

And, as is explained in the comments and in Fulton's book, you construct a toric variety from its fan. If you want to construct a projective toric variety with a set of equations in projective space, there is an explicit way to do that by refining the fan to an integer polytope. But I'm guessing that the other remark more addressed your question.

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Thanks for this answer. So to go off on a tangent, it seems like if you had another field of char 0 you could do a similar construction looking at normal varieties containing a dense open $k[x_1^{\pm},...,x_n^{\pm}]$. This seems ok even for positive characteristic, but now this connection with fans doesn't quite work. Do people study such things or is it not very interesting to generalize in this way? –  solbap Aug 15 '10 at 17:25
    
I am not certainly not an expert in the algebraic geometry. My understanding is, first, that there is a huge difference between having a dense torus and having a torus action with a dense orbit. The latter apparently do generally readily to other fields besides $\mathbb{C}$, and it is interesting in any characteristic, but I don't know a whole lot about that topic. Note though that if $k$ is not algebraically closed, you would want to look at schemes over $k$, and thus implicitly the algebraic closure. –  Greg Kuperberg Aug 15 '10 at 20:59
    
I wonder if anyone have done toric varieties with a non-split torus (over a non-algebraically closed field). –  Lev Borisov Dec 14 '13 at 0:54
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