The Geometry of $T$-Varieties

The present paper is a survey about complex $T$-varieties, i.e. about normal ($n$-dimensional) varieties $X$ over $\mathbb{C}$ with the effective action of a torus $T:=(\mathbb{C}^{*})^{k}$. The case $k=1$ is a classical one — especially singularities with a so-called “good” $\mathbb{C}^{*}$-action have been studied intensively by Pinkham [Pin74, Pin77, Pin78]. The case $n=k$ is classical as well — first studied by Demazure [Dem70], such varieties are called “toric varieties”. The basic theory encodes the category of toric varieties and equivariant morphisms in purely combinatorial terms. Many famous theories in algebraic geometry have their combinatorial counterpart and thus illustrate the geometry from an alternative point of view. The dictionary between algebraic geometry and combinatorics sometimes even helps to establish new theories. The most prominent example of this phenomenon might be the development of mirror symmetry in the 1990’s. Building on the notion of reflexive polytopes, Batyrev gave a first systematic mathematical treatment of this subject [Bat94].

For some features, however, it is useful to consider lower dimensional torus actions as well. For instance, when deforming a toric variety, the embedded torus $T$ still acts naturally on the total- and base-spaces $\widetilde{X}\to S$, but it is usually too small to provide a toric structure on them. The adjacent fibers $X_{s}$ are even worse: depending on the isotropy group of $T$ at the point $s\in S$, only a subtorus of $T$ still acts. Thus, it is worth to study the general case as well. The difference $n-k$ is then called the complexity of a $T$-variety $X$.
While complexity $0$ means toric, the next case of complexity $1$ was systematically studied by Timashev (even for more general algebraic groups) in [Tim97]. On the other hand, Flenner and Zaidenberg [FZ03] gave a very useful description of $\mathbb{C}^{*}$-surfaces even for “non-good” actions. Starting in 2003, there is a series of papers dealing with a general treatment of $T$-varieties in terms of so-called polyhedral divisors. The idea is to catch the non-combinatorial part of a $T$-variety $X$ in an $(n-k)$-dimensional variety $Y$ which is a sort of quotient $Y=X/T$. Now, $X$ can be described by presenting a “polyhedral” divisor $\mathcal{D}$ on $Y$ with coefficients being not numbers but instead convex polyhedra in the vector space $N_{\mathbb{Q}}:=N\otimes_{\mathbb{Z}}\mathbb{Q}$ where $N$ is the lattice of one-parameter subgroups of $T$.

The idea of the present paper is to give an introduction to this subject and to serve as a survey for the many recent papers on $T$-varieties. Moreover, since the notion of polyhedral divisors and the theory of $T$-varieties closely follows the concept of toric varieties, we will treat both cases in parallel. This means that the present paper also serves as a quick introduction to the fascinating field of toric varieties. For a broader discussion and detailed proofs of facts related to toric geometry which are merely mentioned here, the reader is asked to consult any of the standard textbooks like [KKMSD73], [Dan78], [Ful93], and [Oda88]. The subjects of non-toric $T$-varieties are covered in [AH03], [AH06], [AHS08], [AH08], [IS10], [Süß], [PS], [IS], [IVa], [HI], [Vol10], [HS10], [AW], and [AP].

There are also applications of the theory of $T$-varieties and polyhedral divisors in affine geometry [Liea, Lieb, Liec] and coding theory [IS10] which are not covered by this survey. Very recently, $\operatorname{SL}_{2}$-actions on affine T-varieties have also been studied [AL].

Let $M$ and $N$ be two mutually dual, free abelian groups of rank $k$. In other words, both are isomorphic to $\mathbb{Z}^{k}$, and we have a natural perfect pairing $M\times N\to\mathbb{Z}$. Then $T:=\operatorname{Spec}\mathbb{C}[M]=N\otimes_{\mathbb{Z}}\mathbb{C}^{*}$ is the coordinate free version of the torus $(\mathbb{C}^{*})^{k}$ mentioned in (1.1). On the other hand, $N=\mbox{\rm Hom}_{\operatorname{algGr}}(\mathbb{C}^{*},T)$ is the set of one-parameter subgroups of $T$, and $M=\mbox{\rm Hom}_{\operatorname{algGr}}(T,\mathbb{C}^{*})$ equals the character group of $T$. We denote by $M_{\mathbb{Q}}:=M\otimes_{\mathbb{Z}}\mathbb{Q}$ and $N_{\mathbb{Q}}:=N\otimes_{\mathbb{Z}}\mathbb{Q}$ the associated $\mathbb{Q}$-vector spaces.
We always assume that the generators $a^{i}$ for a polyhedral cone $\sigma=\langle a^{1},\ldots,a^{m}\rangle:=\sum_{i=1}^{m}\mathbb{Q}_{\geq 0}\,a^{i}\subseteq N_{\mathbb{Q}}$ are primitive elements of $N$, i.e. they are not proper multiples of other elements of $N$. Dropping this assumption is the first step towards the theory of toric stacks which we will not pursue here, cf. [BCS05]. Moreover, we will often identify a primitive element $a\in N$ with the ray $\mathbb{Q}_{\geq 0}\cdot a$ it generates. Given $\sigma$ as above, we define its dual cone as $\sigma^{\scriptscriptstyle\vee}:=\{u\in M_{\mathbb{Q}}\,|\;\langle\sigma,u\rangle\geq 0\}$. If $\sigma$ does not contain a non-trivial linear subspace (i.e. $0\in\sigma$ is a vertex), then $\dim\sigma^{\scriptscriptstyle\vee}=k$, and we use the semigroup algebra $\mathbb{C}[\sigma^{\scriptscriptstyle\vee}\cap M]:=\oplus_{u\in\sigma^{\scriptscriptstyle\vee}\cap M}\mathbb{C}\,\chi^{u}$ to define the $k$-dimensional toric variety

$$\operatorname{TV}(\sigma):=\operatorname{TV}(\sigma,N):=\operatorname{Spec}\mathbb{C}[\sigma^{\scriptscriptstyle\vee}\cap M].$$

This is a normal variety. If $\dim\sigma=k$, the (finite) set $E\subseteq\sigma^{\scriptscriptstyle\vee}\cap M$ of indecomposable elements is called the Hilbert basis of $\sigma^{\scriptscriptstyle\vee}$. Furthermore, $\operatorname{TV}(\sigma)\subset\mathbb{C}^{E}$ is defined by a binomial ideal which arises from the relations between the elements of the Hilbert basis $E$, cf. (3.2).

The previous notion allows for a generalization. If $\Delta\subseteq N_{\mathbb{Q}}$ is a polyhedron, then we denote by

$$\operatorname{tail}(\Delta):=\{a\in N_{\mathbb{Q}}\,|\;a+\Delta\subseteq\Delta\}$$

its so-called tailcone. Assume that this cone is pointed, i.e. contains no non-trivial linear subspace; then $\Delta$ has a non-empty set of vertices ${\mathcal{V}}(\Delta)$. Denoting their compact convex hull by $\Delta^{c}$, the polyhedron $\Delta$ splits into the “Minkowski” sum $\Delta^{c}+\operatorname{tail}(\Delta)$. Moreover, $\Delta$ gives rise to its inner normal fan $\mathcal{N}(\Delta)$ consisting of the linearity regions of the function $\min\langle\Delta,{\scriptstyle\bullet}\rangle:\operatorname{tail}(\Delta)^{\scriptscriptstyle\vee}\to\mathbb{Q}$. Note that the cones of $\mathcal{N}(\Delta)$ are in a one-to-one correspondence to the faces $F\leq\Delta$ via the map $F\mapsto\mathcal{N}(F,\Delta):=\{u\in M_{\mathbb{Q}}\,|\;\langle F,u\rangle=\min\langle\Delta,u\rangle\}$. We can now define

$$\operatorname{TV}(\Delta):=\operatorname{Spec}\mathbb{C}[\mathcal{N}(\Delta)\cap M]$$

where $\mathbb{C}[\mathcal{N}(\Delta)\cap M]:=\mathbb{C}[\sigma^{\scriptscriptstyle\vee}\cap M]$ as a $\mathbb{C}$-vector space. Note, however, that the multiplication is given by

$$\chi^{u}\cdot\chi^{u^{\prime}}:=\left\{\begin{array}[]{ll}\chi^{u+u^{\prime}}&\mbox{if $u,u^{\prime}$ belong to a common cone of $\mathcal{N}(\Delta)$}\\ 0&\mbox{otherwise}.\end{array}\right.$$

A scheme of the form $\operatorname{TV}(\Delta)$ is called a toric bouquet. We obtain the decomposition

$$\operatorname{TV}(\Delta)=\bigcup_{v\in{\mathcal{V}}(\Delta)}\operatorname{TV}(\mathbb{Q}_{\geq 0}\cdot(\Delta-v))$$

into irreducible (toric) components, compare with (6.1). Note that there is a one-to-one correspondence between polyhedral cones and affine toric varieties, whereas the construction of a toric bouquet only depends on the normal fan $\mathcal{N}(\Delta)$. Hence, dilations of the compact edges of $\Delta$ and translations do not affect the structure of the corresponding bouquet.

Let $M$, $N$, $\sigma\subseteq N_{\mathbb{Q}}$ be as in (2.1). In particular, we assume that $\sigma$ contains no non-trivial linear subspace. These data then give rise to a semigroup (with respect to Minkowski addition)

$$\operatorname{Pol}^{+}_{\mathbb{Q}}(N,\sigma):=\{\Delta\subseteq N_{\mathbb{Q}}\,|\;\Delta\mbox{ polyhedron with }\operatorname{tail}(\Delta)=\sigma\}.$$

Note that $\operatorname{Pol}^{+}_{\mathbb{Q}}(N,\sigma)$ satisfies the cancellation law and contains $\sigma$ as its neutral element. If, in addition, $Y$ is a normal, projective variety over $\mathbb{C}$, then we denote by $\operatorname{CaDiv}_{\geq 0}(Y)$ the semigroup of effective Cartier divisors and call elements

$$\mathcal{D}=\sum_{Z}\mathcal{D}_{Z}\otimes Z\in\operatorname{Pol}^{+}_{\mathbb{Q}}(N,\sigma)\otimes_{\mathbb{Z}_{\geq 0}}\operatorname{CaDiv}_{\geq 0}(Y)$$

polyhedral divisors on $Y$ in $N$ (or simply on $(Y,N)$) with $\operatorname{tail}(\mathcal{D}):=\sigma$. We will also allow $\emptyset$ as an element of $\operatorname{Pol}^{+}_{\mathbb{Q}}(N,\sigma)$ which satisfies $\emptyset+\Delta:=\emptyset$. This somewhat bizarre coefficient is needed to deal with non-compact, open loci defined as $\operatorname{Loc}(\mathcal{D}):=Y\setminus\bigcup_{\mathcal{D}_{Z}=\emptyset}Z$. For each $u\in(\operatorname{tail}(\mathcal{D}))^{\scriptscriptstyle\vee}$, we may then consider the evaluation

$$\mathcal{D}(u):=\sum_{\mathcal{D}_{Z}\neq\emptyset}\min\langle\mathcal{D}_{Z},u\rangle\cdot Z|_{\operatorname{Loc}\mathcal{D}}\in\operatorname{CaDiv}_{\mathbb{Q}}(\operatorname{Loc}\mathcal{D}).$$

This is an ordinary $\mathbb{Q}$-divisor with $\operatorname{supp}\mathcal{D}(u)\subseteq\operatorname{supp}\mathcal{D}:=\operatorname{Loc}\mathcal{D}\cap\bigcup_{\mathcal{D}_{Z}\neq\emptyset}Z$.

Polyhedral divisors have the property that $\mathcal{D}(u)+\mathcal{D}(u^{\prime})\leq\mathcal{D}(u+u^{\prime})$. Hence, they lead to a sheaf of rings ${\mathcal{O}}_{\operatorname{Loc}}(\mathcal{D}):=\bigoplus_{u\in\sigma^{\vee}\cap M}{\mathcal{O}}_{\operatorname{Loc}}(\mathcal{D}(u))\,\chi^{u}$ giving rise to the schemes

$$\operatorname{\widetilde{TV}}(\mathcal{D}):=\operatorname{Spec}_{\operatorname{Loc}(\mathcal{D})}{\mathcal{O}}(\mathcal{D})\mbox{ and the affine }\operatorname{TV}(\mathcal{D}):=\operatorname{Spec}\Gamma(\operatorname{Loc}(\mathcal{D}),{\mathcal{O}}(\mathcal{D})).$$

The latter space does not change if $\mathcal{D}$ is pulled back via a birational modification $Y^{\prime}\to Y$ or if $\mathcal{D}$ is altered by a principal polyhedral divisor on $Y$, that is, an element in the image of the natural map $N\otimes_{\mathbb{Z}}\mathbb{C}(Y)^{*}\to\operatorname{Pol}_{\mathbb{Q}}(N,\sigma)\otimes_{\mathbb{Z}}\operatorname{CaDiv}(Y)$. Two p-divisors which differ by chains of those operations are called equivalent. Note that this implies that $Y$ can always be replaced by a log-resolution.

Consider some $T$-variety $X$ and a $T^{\prime}$-variety $X^{\prime}$. Then the product $X\times X^{\prime}$ carries the natural structure of a $T\times T^{\prime}$-variety. As we shall see, the combinatorial data describing $X\times X^{\prime}$ arise as a kind of product of the combinatorial data describing $X$ and $X^{\prime}$.

For simplicity, we will only consider the affine case, although the construction easily globalizes. Thus, consider p-divisors $\mathcal{D},\mathcal{D}^{\prime}$ on respectively $Y,Y^{\prime}$, with tailcones $\sigma,\sigma^{\prime}$, respectively. We define the product p-divisor $\mathcal{D}\times\mathcal{D}^{\prime}$ on $Y\times Y^{\prime}$ as follows:

$$\mathcal{D}\times\mathcal{D}^{\prime}=\sum_{Z\subset Y}(\mathcal{D}_{Z}\times\sigma^{\prime})\otimes(Z\times Y^{\prime})+\sum_{Z^{\prime}\subset Y^{\prime}}(\sigma\times\mathcal{D}_{Z^{\prime}}^{\prime})\otimes(Y\times Z^{\prime})$$

We will now see that all constructions from the previous section are functorial. Let us start with the setting given in (2.1). A $\mathbb{Z}$-linear map $F\colon N^{\prime}\to N$ satisfying $F_{\mathbb{Q}}(\sigma^{\prime})\subseteq\sigma$ gives rise to a morphism $\operatorname{TV}(F)\colon\operatorname{TV}(\sigma^{\prime},N^{\prime})\to\operatorname{TV}(\sigma,N)$ of affine toric varieties via $F^{\scriptscriptstyle\vee}(\sigma^{\scriptscriptstyle\vee}\cap M)\subseteq(\sigma^{\prime})^{\scriptscriptstyle\vee}\cap M^{\prime}$. For example, if $E\subseteq\sigma^{\scriptscriptstyle\vee}\cap M$ is a Hilbert basis, then the embedding $\operatorname{TV}(\sigma)\hookrightarrow\mathbb{C}^{E}$ from (2.1) is induced by the map $E\colon N\to\mathbb{Z}^{E}$.

A generalization of the functoriality to the setting of (2.3) also has to take care of the underlying variety $Y$.
Let $(Y^{\prime},\mathcal{D}^{\prime},N^{\prime})$ and $(Y,\mathcal{D},N)$ be p-divisors. Consider now any tuple $(F,\phi,\mathfrak{f})$ consisting of a map $F:N\to N^{\prime}$, a dominant morphism $\phi:Y^{\prime}\to Y$, and an element $\mathfrak{f}=\sum v_{i}\otimes f_{i}\in N\otimes\mathbb{C}(Y^{\prime})^{*}$, satisfying $F_{*}\mathcal{D}^{\prime}\subseteq\varphi^{*}\mathcal{D}+\operatorname{div}(\mathfrak{f})$ (to be checked for all coefficients separately). Here,

$$F_{*}\mathcal{D}^{\prime}:=\sum_{Z^{\prime}}F(\mathcal{D}^{\prime}_{Z^{\prime}})\otimes Z^{\prime}$$

is the push forward of $\mathcal{D}^{\prime}$ by $F$,

$$\varphi^{*}\mathcal{D}:=\sum_{Z}\mathcal{D}_{Z}\otimes\varphi^{*}(Z)$$

is the pull back of $\mathcal{D}$ by $\phi$, and

$$\operatorname{div}(\mathfrak{f})=\sum_{i}(v_{i})\otimes\operatorname{div}(f_{i})$$

is the principal polyhedral divisor associated to $\mathfrak{f}$. Such a tuple provides us with naturally defined morphisms

	$\displaystyle\operatorname{\widetilde{TV}}(F,\varphi,\mathfrak{f})\colon\operatorname{\widetilde{TV}}(\mathcal{D}^{\prime},N^{\prime})\to\operatorname{\widetilde{TV}}(\mathcal{D},N),\qquad\mbox{and}$
	$\displaystyle\operatorname{TV}(F,\varphi,\mathfrak{f})\colon\operatorname{TV}(\mathcal{D}^{\prime},N^{\prime})\to\operatorname{TV}(\mathcal{D},N).$

This construction yields an equivalence of categories between polyhedral cones and affine toric varieties in the toric case (where only the map $F$ plays a role). To obtain a similar result for the relation between p-divisors and $T$-varieties one first needs to define the category of the former by turning both types of equivalences mentioned in (2.3) into isomorphisms. This can be done by the common technique of localization which is known from the construction of derived categories. In the category of $T$-varieties we restrict to morphisms $\psi:X^{\prime}\rightarrow X$, with $T.\psi(X^{\prime})\subset X$ being dense. We will call such morphisms orbit dominating.

Let us fix a torus $T$. To glue affine $T$-varieties together one has to understand $T$-equivariant open embeddings. In the toric case, an inclusion $F_{\mathbb{Q}}:\sigma^{\prime}\hookrightarrow\sigma$ of cones in $N_{\mathbb{Q}}$ (i.e. $F=\operatorname{id}_{N}$ with $\sigma^{\prime}\subseteq\sigma$) provides an open embedding $\operatorname{TV}(F)$ if and only if $\sigma^{\prime}$ is a face of $\sigma$. Indeed, if $\sigma^{\prime}=\operatorname{face}(\sigma,u):=\sigma\cap u^{\bot}$ is cut out by the supporting hyperplane $u\in\sigma^{\scriptscriptstyle\vee}$, then $\mathbb{C}[{\sigma^{\prime}}^{\scriptscriptstyle\vee}\cap M]=\mathbb{C}[\sigma^{\scriptscriptstyle\vee}\cap M]_{\chi^{u}}$ equals the localization by $\chi^{u}$. Thus, $\operatorname{TV}(\sigma^{\prime})=[\chi^{u}\neq 0]\subseteq\operatorname{TV}(\sigma)$.

Similarly, if $\mathcal{D}$ is a p-divisor on $Y$ and $f\in{\mathcal{O}}_{\operatorname{Loc}\mathcal{D}}(\mathcal{D}(u))$, then $f(y)\chi^{u}\in\mathbb{C}(Y)[M]\subseteq\mathbb{C}(\operatorname{TV}(\mathcal{D}))$ is an $M$-homogeneous rational function on $\operatorname{TV}(\mathcal{D})$. The localization procedure of the toric setting now generalizes to to the present setting: the open subset $[f\chi^{u}\neq 0]\subseteq\operatorname{TV}(\mathcal{D})$ is again a $T$-variety. Its associated p-divisor $\mathcal{D}_{f\chi^{u}}$ is still supported on $Y$ and its tailcone is $\operatorname{tail}(\mathcal{D}_{f\chi^{u}})=\operatorname{face}(\operatorname{tail}(\mathcal{D}),u)$. Moreover,

$$\mathcal{D}_{f\chi^{u}}=\operatorname{face}(\mathcal{D},u)+\emptyset\otimes(\operatorname{div}f+\mathcal{D}(u))$$

where the face operator is supposed to be applied to the polyhedral coefficients only. More specifically, $\operatorname{face}(\Delta,u):=\{a\in\Delta\,|\;\langle a,u\rangle=\min\langle\Delta,u\rangle\}$. Note that $Z_{u}(f):=\operatorname{div}f+\mathcal{D}(u)$ is an effective divisor which depends “continuously” upon $f$.
In contrast to the toric case, not all $T$-invariant open subsets are of the form $[f\chi^{u}\neq 0]$. We will now present a precise characterization of open embeddings, which is quite technical, but apparently does not easily simplify in the general case. For the much nicer case of complexity one, however, we refer to (5.2). We need the following notation: for a polyhedral divisor $\mathcal{D}=\sum_{Z}\mathcal{D}_{Z}\otimes Z$ and a not necessarily closed point $y\in Y$ we define $\mathcal{D}_{y}:=\sum_{y\in Z}\mathcal{D}_{Z}\in\operatorname{Pol}^{+}_{\mathbb{Q}}(N,\sigma)$.
For any two p-divisors $\mathcal{D},\mathcal{D}^{\prime}$ on a common $(Y,N)$, we say that $\mathcal{D}^{\prime}\subseteq\mathcal{D}$ if the polyhedral coefficients respectively satisfy $\mathcal{D}^{\prime}_{Z}\subseteq\mathcal{D}_{Z}$. Two such p-divisors induce maps $\widetilde{\iota}:\operatorname{\widetilde{TV}}(\mathcal{D}^{\prime})\to\operatorname{\widetilde{TV}}(\mathcal{D})$ and $\iota:\operatorname{TV}(\mathcal{D}^{\prime})\to\operatorname{TV}(\mathcal{D})$. The former is an open embedding if and only if $\mathcal{D}^{\prime}\leq\mathcal{D}$, i.e. if and only if all coefficients $\mathcal{D}^{\prime}_{Z}$ are faces of the corresponding $\mathcal{D}_{Z}$. In particular, this implies that $\operatorname{tail}(\mathcal{D}^{\prime})\leq\operatorname{tail}(\mathcal{D})$.

The characterization of open embeddings for affine toric varieties via the face relation between polyhedral cones leads directly to the notion of a (polyhedral) fan which is a special instance of a polyhedral subdivision:

Similar to the toric case, we can construct a scheme from a divisorial fan $\mathcal{S}$ via

$$\operatorname{TV}(\mathcal{S}):=\bigcup_{\mathcal{D}\in\mathcal{S}}\operatorname{TV}(\mathcal{D})\hskip 8.00003pt\mbox{with}\hskip 8.00003pt\operatorname{TV}(\mathcal{D})\cap\operatorname{TV}(\mathcal{D}^{\prime})=\operatorname{TV}(\mathcal{D}\cap\mathcal{D}^{\prime}).$$

In general, the resulting scheme may not be separated, but there is a combinatorial criterion for checking this, see [AHS08, Theorem 7.5]. On the other hand, by covering any $T$-variety $X$ with invariant affine open subsets, it is possible to construct a divisorial fan $\mathcal{S}$ with $X=\operatorname{TV}(\mathcal{S})$, see [AHS08, Theorem 5.6].

In general, the notion of divisorial fan can be cumbersome to work with, in large part due to the technical nature of Proposition 9. However, a manageable substitute can be obtained in complexity 1, see (5.3).

The affine toric setting presented in (2.1) fits into the language of (2.3) when setting $Y=\operatorname{Spec}\mathbb{C}$. Since divisors on points are void, the only information carried by $\mathcal{D}$ is its tailcone $\sigma$. Thus, $\operatorname{\widetilde{TV}}(\mathcal{D})=\operatorname{TV}(\mathcal{D})=\operatorname{TV}(\sigma)$ in this case.

The toric case, however, can also provide us with more interesting examples. To this end we consider a toric variety $\operatorname{TV}(\delta,\widetilde{N})$ and fix a subtorus action $T\hookrightarrow\widetilde{T}$. Now, what is the description of $\operatorname{TV}(\delta,\widetilde{N})$ as a $T$-variety? Assuming that the embedding $T\hookrightarrow\widetilde{T}$ is induced from a surjection of the corresponding character groups $p:\widetilde{M}\rightarrow\hskip-8.00003pt\rightarrow M$, we denote the kernel by $M_{Y}$ and obtain two mutually dual exact sequences (7) and (12).
Setting $\sigma:=N_{\mathbb{Q}}\cap\delta$, the map $p$ gives us a surjection $\delta^{\scriptscriptstyle\vee}\rightarrow\hskip-8.00003pt\rightarrow\sigma^{\scriptscriptstyle\vee}$. On the dual side, denote the surjection $\widetilde{N}\rightarrow\hskip-8.00003pt\rightarrow N_{Y}$ by $q$. However, instead of just considering the cone $q(\delta)$, we denote by $\Sigma$ the coarsest fan refining the images of all faces of $\delta$ under the map $q$. Then, $|\Sigma|=q(\delta)$, and the set $\Sigma(1)$ of rays in $\Sigma$, i.e. its one-dimensional cones, contains the set $q(\delta(1))$.

(7)
(12)

Next, we perform the following two mutually dual constructions. For $u\in\sigma^{\scriptscriptstyle\vee}$ and $a\in|\Sigma|$, let $\nabla(u)\subseteq M_{Y,\mathbb{Q}}$ and $\mathcal{D}_{a}\subseteq N_{\mathbb{Q}}$ be as indicated in the diagram above. Note that we have to make use of some section $s:M\hookrightarrow\widetilde{M}$ of $p$, i.e. a splitting of the two sequences, to shift both polyhedra from $p^{-1}(u)$ and $q^{-1}(a)$ into the respective fibers over $0$. Both polyhedra are linked to each other by the equality

$$\min\langle a,\nabla(u)\rangle=\min\langle\mathcal{D}_{a},u\rangle$$

which is an easy consequence (via $a=q(x)$ and $u=p(y)$) of the following lemma appearing in the proof of [AH03, Proposition 8.5].

Thus, defining $Y:=\operatorname{TV}(\Sigma)$, we can refer to the upcoming discussion in (6.1) for the construction of $T$-invariant Weil divisors $\overline{\operatorname{orb}}(a)\subseteq\operatorname{TV}(\Sigma)$ for $a\in\Sigma(1)$ to define the p-divisor $\mathcal{D}^{\delta}:=\sum_{a\in\Sigma(1)}\mathcal{D}_{a}\otimes\overline{\operatorname{orb}}(a)$ on $(Y,N)$. Although we have indexed the coeffixients of $\mathcal{D}$ by a ray $a$ instead of a prime divisor, this is harmless since we may always identify $a$ with the divisor $\overline{\operatorname{orb}}(a)\subseteq\operatorname{TV}(\Sigma)$. It follows from the characterization of toric nef Cartier divisors and the above equality involving $\nabla(u)$ and $\mathcal{D}_{a}$ that $\operatorname{TV}(\delta)=\operatorname{TV}(\mathcal{D}^{\delta})$.

Let $X\subseteq\mathbb{C}^{n}$ be an equivariantly embedded, normal, affine $T$-variety which intersects the torus non-trivially, and denote by $f_{1},\ldots,f_{m}$ $M$-homogeneous generating equations for $X$. If $T$ acts diagonally and effectively on $\mathbb{C}^{n}$, then each variable $x_{i}$ has a degree in $M$ which gives rise to a surjection $p:\widetilde{M}:=\mathbb{Z}^{n}\rightarrow\hskip-8.00003pt\rightarrow M$, $e_{\nu}\mapsto\deg x_{\nu}$ as in the previous section. Setting $\delta:=\mathbb{Q}^{n}_{\geq 0}$ and $u_{i}:=\deg f_{i}\in M$, we use the section $s:M\hookrightarrow\widetilde{M}$ as mentioned in (4.2) to shift the equations $f_{i}$ to the Laurent polynomials $f^{\prime}_{i}:=\chi^{-s(u_{i})}f_{i}\in\mathbb{C}[M_{Y}]$. They define a closed subvariety of the torus $T_{Y}:=\operatorname{Spec}\mathbb{C}[M_{Y}]$, and we use this to modify the definition of $Y$ from (4.2) to the normalization of

$$\overline{V(f^{\prime}_{1},\ldots,f^{\prime}_{m})}\subseteq\operatorname{TV}(\Sigma).$$

If $\overline{\operatorname{orb}}(a)$ are Cartier divisors, then we can consider their pull backs $Z_{a}:=\overline{\operatorname{orb}}(a)\cap Y$. Thus, we arrive at the description $X=\operatorname{TV}(\mathcal{D})$ as a $T$-variety, where $\mathcal{D}=\sum_{a\in\Sigma(1)}\mathcal{D}_{a}\otimes Z_{a}$.

At the end of (3.4) we introduced the notion of divisorial fans, which encode invariant open affine coverings of general $T$-varieties. The face conditions guarantees that for any divisorial fan $\mathcal{S}=\{\mathcal{D}^{i}\}$ and any prime divisor $Z\subset Y$, the set

$$\mathcal{S}_{Z}=\{\mathcal{D}_{Z}^{i}\}$$

is a polyhedral subdivision in $N_{\mathbb{Q}}$ called a slice. Indeed, since open embeddings between affine $T$-varieties $\operatorname{TV}(\mathcal{D}^{\prime})\hookrightarrow\operatorname{TV}(\mathcal{D})$ imply that $\mathcal{D}^{\prime}\leq\mathcal{D}$, the polyhedra $\mathcal{D}_{Z}^{i}\subseteq N_{\mathbb{Q}}$ are supposed to be glued along the common faces $\mathcal{D}^{\prime}_{Z}\leq\mathcal{D}_{Z}$.

These slices can be viewed another way as well. Similarly to (4.3), one can try to embed a general $T$-variety $X$ into a toric variety $\operatorname{TV}(\mathcal{F})$ with some fan $\mathcal{F}$ of polyhedral cones $\delta$ in $\widetilde{N}_{\mathbb{Q}}$. Now, the method of (4.2) can be copied to yield an $\mathcal{S}$ on $\operatorname{TV}(\Sigma)$ describing $\operatorname{TV}(\mathcal{F})$ as a $T$-variety denoted by $\operatorname{TV}(\mathcal{S})$ and, afterwards, one follows (4.3) to find the right $Y\subseteq\operatorname{TV}(\Sigma)$ to which $\mathcal{S}$ should be restricted in order to describe $X$. In this setting, the slices $\mathcal{S}_{Z}$ appear as intersections of the fan $\mathcal{F}$ with certain affine subspaces, explaining the choice of terminology.
It is tempting to think that the slices $\mathcal{S}_{Z}$ capture the entire information of $\mathcal{S}$. However, the important point is that one also must keep track of which of the polyhedral cells inside the different $\mathcal{D}_{Z}$ belong to a common p-divisor. This is related to the question of how much is contracting along $\operatorname{\widetilde{TV}}(\mathcal{S})\to\operatorname{TV}(\mathcal{S})$. Thus, one is supposed to introduce labels for the cells in $\mathcal{S}_{Z}$. Again, we refer to (5.3) for a nice way of overcoming this problem in complexity one.

Let $\mathcal{D}=\sum_{P\in Y}\mathcal{D}_{P}\otimes P$ denote a p-divisor of complexity one on the curve $Y$ with tailcone $\sigma$. Since prime divisors $Z$ coincide with closed points $P\in Y$ we will from now on replace the letter $Z$ by $P$ in this setting. As in the definition of $\mathcal{D}_{P}$ in (3.3) we can now introduce the degree

$$\deg\mathcal{D}:=\sum_{P\in Y}\mathcal{D}_{P}\in\operatorname{Pol}^{+}_{\mathbb{Q}}(N,\sigma).$$

Note that it satisfies the equation $\min\langle\deg\mathcal{D},u\rangle=\deg\mathcal{D}(u)$. In contrast, considering the sum of the polyhedral coefficients of a p-divisor in higher complexity does not make much sense. Nonetheless, one may define the degree of a p-divisor on a polarized $Y$ via the following construction: let $C\subseteq Y$ be a curve or, more generally, a numerical class of curves in $Y$. Then there is a well defined generalized polyhedron $(C\cdot\mathcal{D})\in\operatorname{Pol}_{\mathbb{Q}}(N,\sigma)$ where the term “generalized” refers to an element in the Grothendieck group associated to $\operatorname{Pol}^{+}_{\mathbb{Q}}(N,\sigma)$. Hence, it is representable as the formal difference of two polyhedra.

For a p-divisor $\mathcal{D}$ on a curve $Y$ it is as obvious as important to observe that

$$\deg\mathcal{D}\neq\emptyset\;\Leftrightarrow\;\operatorname{Loc}\mathcal{D}=Y.$$

Moreover, if $\mathcal{D}$ is a polyhedral divisor on $Y$ then it is not hard to see that $\mathcal{D}$ is a p-divisor $\Leftrightarrow$ $\deg\mathcal{D}\subsetneq\operatorname{tail}(\mathcal{D})$ and $\mathcal{D}(w)$ has a principal multiple for all $w\in(\operatorname{tail}(\mathcal{D}))^{\scriptscriptstyle\vee}$ with $w^{\bot}\cap(\deg\mathcal{D})\neq 0$. Note that the latter condition is automatically fulfilled if $Y=\mathbb{P}^{1}$.
Nonetheless, degree polyhedra play their most important roles in the characterization of open embeddings since they transform the technical condition from (3.3) into a very natural and geometric one.

In particular, if $\mathcal{S}=\{\mathcal{D}^{i}\}$ is a divisorial fan as in (3.4) then the set $\{\operatorname{tail}(\mathcal{D}^{i})\,|\,\mathcal{D}^{i}\in\mathcal{S}\}$ forms the so-called tailfan $\operatorname{tail}(\mathcal{S})$, and the subsets $\deg\mathcal{D}^{i}\subsetneq\operatorname{tail}(\mathcal{D}^{i})$ glue together to a proper subset $\deg\mathcal{S}\subsetneq|\operatorname{tail}(\mathcal{S})|$. The single degree polyhedra can be recovered as $\deg\mathcal{D}^{i}=\deg\mathcal{S}\cap\operatorname{tail}(\mathcal{D}^{i})$.

Another important feature of curves as base spaces $Y$ is that $\operatorname{Loc}\mathcal{D}\subseteq Y$ is either projective (if $\deg\mathcal{D}\neq\emptyset$) or affine (if $\deg\mathcal{D}=\emptyset$). In the latter case, $\operatorname{\widetilde{TV}}(\mathcal{D})=\operatorname{TV}(\mathcal{D})$, and gluing those cells does not cause problems. For the other case, an easy observation is

Indeed, we have the following chain of inequalities

$$\dim(\operatorname{tail}(\mathcal{D}))\leq\dim\mathcal{D}_{P}\leq\dim(\deg\mathcal{D})\leq\dim(\operatorname{tail}(\mathcal{D}))$$

each of which results from a combination of translations and inclusions.

Indeed, we can take the divisorial fan $\mathcal{S}$ to consist of p-divisors $\mathcal{D}^{\sigma}$ for full-dimensional $\sigma\in\operatorname{tail}(\mathcal{S})$ with non-empty degree, for $\sigma$ with empty degree any finite set of p-divisors providing an open affine covering of $\operatorname{\widetilde{TV}}(\mathcal{D}^{\sigma})$, and intersections thereof.

If $\mathcal{S}$ is an f-divisor, i.e. a pair $(\sum_{P}\mathcal{S}_{P}\otimes P,\mathfrak{deg})$, we write $\operatorname{TV}(\mathcal{S})$ for the $T$-variety it determines as in proposition 20. Note that we use the same symbol for divisorial fans and f-divisors, since they both determine general $T$-varieties (although the former also contains the information of a specific affine cover). As has already been observed in the affine case of (2.3), different f-divisors $\mathcal{S},\mathcal{S}^{\prime}$ might yield the same (or equivariantly isomorphic) $T$-varieties $\operatorname{TV}(\mathcal{S})=\operatorname{TV}(\mathcal{S}^{\prime})$. This is the case if there are isomorphisms $\varphi:Y\rightarrow Y^{\prime}$ and $F:N\rightarrow N^{\prime}$ and an element $\sum_{i}v_{i}\otimes f_{i}\in N\otimes\mathbb{C}(Y^{\prime})$ such that $\mathcal{S}^{\prime}_{\varphi(P)}+\sum_{i}\operatorname{ord}_{P}(f_{i})\cdot v_{i}=F(\mathcal{S}_{P})$ holds for every $P\in Y$ and we have $\mathfrak{deg}^{\prime}=F(\mathfrak{deg})$.