I am reading Toric Varieties by Cox, Little, and Schenck. I am stuck on the definition an algebraic torus, given in Part 1.1, page 10, which states:

The affine variety $(\mathbb{C}^*)^n$ is a group under componentwise multiplication. A torus $T$ is an affine variety isomorphic to $(\mathbb{C}^*)^n$, where $T$ inherits a group structure from the isomorphism.

Now, my understanding of the definition of an affine variety, as per Hartshorne is a subset $A$ of affine $n$-space over an algebraically closed field $k$, such that $A = Z(B)$, i.e. $A$ is the zero-locus of some set $B\subset k[X_1,\dots,X_n]$.

My question is, why is it stated that $(\mathbb{C}^*)^n$ is an affine variety? What set of polynomials is it the zero-locus of?

In particular, if $n=1$, then the only algebraic sets are finite sets, so how can $\mathbb{C}^* = \mathbb{C}\setminus \{0\}$ be an algebraic set if it is infinite?

Now, I suspect that affine variety should maybe be just variety as per Hartshorne's definition in Hartshorne page 15 where he defined a variety to be any affine, quasi-affine, projective, or quasi-projective variety. And thus together with morphisms between varieties we have the category of varieties.

Also I talked to some other people who know more about algebraic geometry who seems to not understand why there is confusion, since apparently they have a more general and abstract definition of a variety than merely as zero-loci. Whereas I understand that maybe after I've read enough of Hartshorne I may resolve this issue, the prerequisites listed for this particular chapter of Toric Varieties, which is also another book by Cox, called Ideals, Varieties, and Algorithms also only uses this "elementary" definition. Therefore I expect this question to be resolvable within the bounds of my current progress of Hartshorne.

Would glad to have any help, thanks.

  • 5
    $\begingroup$ For $n = 1$, for example, consider $\mathbb{C}^* = \{(x, y)\in \mathbb{C}^2:\, xy = 1\}$. $\endgroup$ – anomaly Mar 4 at 15:15
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    $\begingroup$ For $K$ algebraically closed $K^*$ is affine in the sense that $K^* \to Z(xy-1), x \mapsto (x,1/x)$ is a birational map whose poles are outside of the domains. When considering $K= \mathbb{C}$ you may allow biholomorphic maps instead of just birational (so that $\mathbb{C}/2i \pi \mathbb{Z} \cong \mathbb{C}^*$) $\endgroup$ – reuns Mar 4 at 15:33

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