Definition of Toric Varieties

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

The affine toric variety defined in §Affine Toric Varieties, ⁋Definition 5 is the algebraic variety $U_\tau$ corresponding to a single strongly convex rational polyhedral cone $\tau$. As we saw in §Affine Toric Varieties, ⁋Example 14, this can be regarded as an algebraic variety equipped with a torus action structure, and by definition it is the $\Spec$ of a suitable ring, hence an affine variety.

In this post we examine how a general toric variety is obtained by gluing affine toric varieties. Specifically, we saw that the affine toric variety $U_\tau$ corresponding to a face $\tau$ of a cone $\sigma$ is a principal open subset of $U_\sigma$ (§Affine Toric Varieties, ⁋Proposition 13); equivalently, if two cones share a common face, this yields inclusions between the affine toric varieties they define, and thus allows us to perform gluing.

Definition of a Fan

To glue affine toric varieties together as above, we need a combinatorial structure controlling how the individual cones meet one another. For this purpose we define a fan.

Definition 1 Given a lattice $N$, a fan $\Sigma$ in $N_\mathbb{R}$ is a collection of strongly convex rational polyhedral cones satisfying the following conditions:

Every face of a cone in $\Sigma$ also belongs to $\Sigma$.
The intersection of any two cones in $\Sigma$ is a face of each.

The second condition is what makes the gluing process described above work in Definition 3; it forces two distinct cones $\tau_1, \tau_2$ to meet only along their common face. The first condition, meanwhile, can be viewed as a kind of closedness: since a fan contains all faces of each of its cones.

Example 2 In $N = \mathbb{Z}^2$, we can consider a fan consisting of three 2-dimensional cones $\tau_0, \tau_1, \tau_2$ arranged radially about the origin and covering $\mathbb{R}^2$. For instance, take the three vectors $e_1, e_2, -e_1-e_2$ and consider the three cones they generate.

$The fan of $$\mathbb{P}^2$$$

Each cone is generated by two ray generators $\rho_i, \rho_{i+1}$, and these rays become the 1-dimensional cones. The origin $\{0\}$ itself is a 0-dimensional cone contained in every fan. This fan is the most basic example defining the toric variety of $\mathbb{P}^2$.

Definition of a Toric Variety

Now, given a fan $\Sigma$, we can define an algebraic variety $X_\Sigma$ from it. In fact, all the basic tools needed for this have already been explained; only the language of Definition 1 was additionally required. To summarize the construction in precise mathematical terms: given a fan $\Sigma$, for each cone $\tau \in \Sigma$ we obtain the affine toric variety $U_\tau$ by §Affine Toric Varieties, ⁋Definition 5. Then, when two cones $\tau_1, \tau_2 \in \Sigma$ have a common face $\tau_1 \cap \tau_2$, §Affine Toric Varieties, ⁋Proposition 13 implies that $U_{\tau_1 \cap \tau_2}$ is a principal open subset of both $U_{\tau_1}$ and $U_{\tau_2}$, and thus we have isomorphisms and inclusions

\[U_{\tau_1} \supset U_{\tau_1 \cap \tau_2} \cong U_{\tau_2 \cap \tau_1} \subset U_{\tau_2}\]

which allow us to glue the $U_\tau$ together.

Definition 3 Given a fan $\Sigma$, the algebraic variety obtained by gluing the affine toric varieties $\{U_\tau\}_{\tau \in \Sigma}$ in the manner described above is called the toric variety defined by $\Sigma$, and is denoted $X_\Sigma$.

The following generalizes §Affine Toric Varieties, ⁋Proposition 15.

Proposition 4 The toric variety $X_\Sigma$ is a normal, separated algebraic variety.

Proof

The normality of $X_\Sigma$ follows because each affine chart $U_\tau$ is normal by §Affine Toric Varieties, ⁋Proposition 15, and this property is preserved under gluing. To show separatedness, it suffices to verify that the image of the diagonal morphism $\Delta: X_\Sigma \to X_\Sigma \times X_\Sigma$ is closed. On each affine chart this holds by the second condition of a fan, which guarantees that the intersection of two cones is a face.

One important property that the toric variety $X_\Sigma$ inherits from the affine toric varieties is that it contains the algebraic torus $T_N$ as an open dense subset (§Affine Toric Varieties, ⁋Proposition 11). Indeed, the affine chart $U_{\{0\}}$ corresponding to the 0-dimensional cone $\{0\} \in \Sigma$ is isomorphic to $T_N$, and every other $U_\tau$ contains it as an open subset. Hence $T_N \subset X_\Sigma$ defines an open dense embedding.

As we saw in §Affine Toric Varieties, ⁋Proposition 10, each affine toric variety $U_\tau$ carries a natural action of the algebraic torus $T_N$. In that post we observed that the inclusion between affine toric varieties induced by an inclusion of cones is invariant under the torus action, so this action extends to all of $X_\Sigma$ via gluing.

Proposition 5 For any fan $\Sigma$, a natural action of the algebraic torus $T_N$ is defined on the toric variety $X_\Sigma$. Under this action, $T_N \subset X_\Sigma$ is an open dense $T_N$-invariant subset, and the action of $T_N$ on itself extends to the action on $X_\Sigma$.

Proof

For each cone $\tau \in \Sigma$, the $T_N$-action on $U_\tau$ is defined by §Affine Toric Varieties, ⁋Proposition 10. Since these actions agree on the intersection $U_{\tau_1 \cap \tau_2}$ of two affine charts $U_{\tau_1}$ and $U_{\tau_2}$, they define a well-defined $T_N$-action on all of $X_\Sigma$. Meanwhile, the chart $U_{\{0\}} \cong T_N$ corresponding to $\{0\} \in \Sigma$ is an open dense subset, and the self-action of $T_N$ is given by left multiplication on this chart, so it extends naturally to the action on $X_\Sigma$.

If a fan $\Sigma$ satisfies $\bigcup_{\tau \in \Sigma} \tau = N_\mathbb{R}$, we call it a complete fan. In this case $X_\Sigma$ is known to be a complete algebraic variety, i.e., a variety proper over $\Spec(\mathbb{C})$. Since completeness is the algebraic analogue of compactness, in this case we call $X_\Sigma$ an equivariant compactification of $T_N$.

The structure of the orbits under this action is closely related to the combinatorics of the fan. Specifically, the orbit $O(\tau)$ corresponding to a $d$-dimensional cone $\tau \in \Sigma$ is isomorphic to a torus $(\mathbb{C}^\ast)^{n-d}$ of dimension $n-d$; in particular, the orbit corresponding to an $n$-dimensional cone, i.e., a maximal cone, is 0-dimensional, hence a fixed point of the $T_N$-action.

The Normal Fan and Projective Toric Varieties

So far we have examined how to construct a toric variety $X_\Sigma$ from an arbitrary fan $\Sigma$. Now we discuss a special kind of fan: the normal fan naturally associated to a polytope. This is the key construction for producing projective toric varieties.

Let $N$ be the dual lattice of a lattice $M$, and let $P$ be a full-dimensional convex lattice polytope in $M_\mathbb{R}$. For each facet $F^\prime$ of $P$, among the elements of $N$ perpendicular to the hyperplane determined by $F^\prime$ and pointing toward the interior of $P$, the unique lattice vector that cannot be written in the form $v = k v^\prime$ with $k > 1$ an integer and $v^\prime \in N$ is called the primitive inner normal vector $u_{F^\prime} \in N$ of $F^\prime$.

Definition 6 The normal fan $\Sigma_P$ of a polytope $P$ is defined as the collection of cones $\tau_F$, one for each face $F$ of $P$, where $\tau_F$ is generated by the primitive inner normal vectors $u_{F^\prime}$ of all facets $F^\prime$ containing $F$:

\[\Sigma_P = \{\tau_F \mid F \text{ is a face of } P\}\]

Our first claim is that this is indeed a fan, and therefore defines a toric variety.

Proposition 7 The collection $\Sigma_P$ obtained from the definition above is indeed a fan. That is, $\Sigma_P$ satisfies both conditions of Definition 1.

Proof

Any element $\tau_F$ of $\Sigma_P$ is generated by $\mathbb{R}_{\ge 0}$-linear combinations of primitive inner normal vectors, so it is a strongly convex rational polyhedral cone. First, let us verify the face condition. A face of $\tau_F$ corresponds to a face $F^\prime \supseteq F$ containing $F$, and this yields $\tau_{F^\prime} \in \Sigma_P$, so the first condition is satisfied. Consider the intersection of two cones $\tau_{F_1}, \tau_{F_2} \in \Sigma_P$. The intersection $\tau_{F_1} \cap \tau_{F_2}$ equals the cone $\tau_F$ corresponding to the smallest face $F$ containing the intersection $F_1 \cap F_2$. This is a face of both $\tau_{F_1}$ and $\tau_{F_2}$, so the second condition is also satisfied.

Thus the normal fan defines a toric variety $X_{\Sigma_P}$. What we examine next is that the toric variety obtained in this way is not merely a toric variety but a projective variety, and furthermore that projective toric varieties are in fact all obtained in this manner. To describe this correspondence precisely, it is useful to first organize the relationship between line bundles on a toric variety and polytopes.

As we already saw right after Proposition 5, the stratum structure of $X_\Sigma$ corresponds to the cones of the fan $\Sigma$. In particular, the 0-dimensional cone $\{0\}$ corresponds to the open dense torus $T_N \subset X_\Sigma$, and the 1-dimensional cones $\rho \in \Sigma(1)$ (where $\Sigma(1)$ denotes the set of 1-dimensional cones of $\Sigma$) correspond respectively to codimension-1 $T_N$-invariant prime divisors $D_\rho \subset X_\Sigma$. Hence the free abelian group $\bigoplus_\rho \mathbb{Z} D_\rho$ describes all $T_N$-invariant Weil divisors on $X_\Sigma$.

Since we are interested in line bundles on toric varieties, we must focus on Cartier divisors. A detailed discussion describing Cartier divisors in the language of piecewise linear functions and relating them to line bundles will be given in the next post, §Torus-Invariant Divisors and Line Bundles; however, for the purposes of the present discussion we first import the result of §Torus-Invariant Divisors and Line Bundles, ⁋Proposition 6.

According to this, a $T_N$-invariant Weil divisor $D = \sum_\rho a_\rho D_\rho$ is Cartier if and only if for each maximal cone $\sigma \in \Sigma$ there exists some $m_\sigma \in M$ such that $a_\rho = -\langle m_\sigma, v_\rho \rangle$ holds for every ray $\rho$ of $\sigma$. Here $v_\rho$ denotes the primitive generator of $\rho$. For such a Cartier divisor $D$, we define a polytope $P_D \subset M_\mathbb{R}$ by

\[P_D = \{u \in M_\mathbb{R} \mid \langle u, v_\rho \rangle \ge -a_\rho \text{ for all } \rho \in \Sigma(1)\}\]

Then the above compatibility condition corresponds exactly to the vertices of $P_D$ being given by lattice elements $m_\sigma$, and therefore $P_D$ becomes a lattice polytope. Conversely, the process of obtaining the normal fan $\Sigma_P$ from a lattice polytope $P$ is as we saw in Definition 6. This two-way correspondence forms the bridge between the projectivity of a toric variety and polytopes.

Proposition 8 The toric variety $X_\Sigma$ is a projective variety if and only if $\Sigma$ is the normal fan of some full-dimensional lattice polytope $P$, i.e., $\Sigma = \Sigma_P$.

Proof

($\Rightarrow$) If $X_\Sigma$ is projective, then there exists a very ample line bundle $\mathcal{L}$ on it, and by the correspondence above this is expressed as a $T_N$-invariant Cartier divisor $D$. As we saw earlier, the resulting $P_D$ is a lattice polytope, and since $\mathcal{L}$ is ample (in particular very ample), the corresponding piecewise linear function $\psi_D$ is strictly convex by §Torus-Invariant Divisors and Line Bundles, ⁋Proposition 9. From this we obtain that the normal fan of $P_D$ coincides with $\Sigma$.

($\Leftarrow$) Suppose $\Sigma = \Sigma_P$. From the data of the polytope $P$—namely, for each ray $\rho$ the value $a_\rho = -\min_{u \in P}\langle u, v_\rho\rangle$—we obtain the $T_N$-invariant divisor $D_P = \sum_\rho a_\rho D_\rho$. Here the fact that each vertex of $P$ is a lattice point exactly satisfies the preceding compatibility condition, making $D_P$ Cartier, and for sufficiently large $k > 0$, the multiple $kD_P$ becomes very ample. Then the Zariski closure of the image of the monomial map $\phi_{kP}: T_N \to \mathbb{P}^s$ defined by the lattice points of $kP$ is isomorphic to $X_\Sigma$ (see Proposition 9), so $X_\Sigma$ is projective.

A toric variety $X_\Sigma = X_{\Sigma_P}$ satisfying this equivalence condition is called a projective toric variety, and when emphasizing the polytope $P$ it is also written $X_P$. This result is a representative example showing that the geometric properties of a toric variety are completely described by the combinatorial properties of its fan.

Meanwhile, another method of constructing the projective toric variety $X_P$ from a polytope $P$ is to give an explicit embedding via a monomial map. The key idea is that $T_N$ sits inside $X_P$ as an open dense subset, so if we embed $T_N$ appropriately into $\mathbb{P}^s$, the destination of the remaining points is automatically determined.

Consider the lattice points of $P \subset M_\mathbb{R}$, i.e., $P \cap M = \{m_0, m_1, \ldots, m_s\}$. Using these we define the monomial map

\[\phi_P: T_N \longrightarrow \mathbb{P}^s, \qquad t \longmapsto [\rchi^{m_0}(t) : \rchi^{m_1}(t) : \cdots : \rchi^{m_s}(t)]\]

Here $\rchi^m: T_N \to \mathbb{C}^\ast$ is the character corresponding to $m \in M$.

Proposition 9 The Zariski closure of the image of $\phi_P$ defined above is isomorphic to $X_P$. That is, $X_P \cong \overline{\phi_P(T_N)}$.

Proof

When $P$ is a very ample lattice polytope (i.e., when the corresponding divisor $D_P$ on $X_P$ defines a very ample line bundle), the affine chart corresponding to each vertex $v$ has a coordinate ring generated by the monomials $\rchi^{m_i - m_v}$. Here $m_v$ is the lattice point corresponding to the vertex $v$. These affine charts are naturally compatible with the standard affine charts of $\mathbb{P}^s$, and their gluing defines $\overline{\phi_P(T_N)}$. In the general case, choose $k > 0$ such that $kP$ becomes very ample. Since scaling a polytope by a positive integer does not change its normal fan, we have $\Sigma_{kP} = \Sigma_P$ and therefore $X_{kP} = X_P$, and the embedding is defined as above using $kP$.

This embedding once again illustrates the link between the combinatorial definition and the algebro-geometric definition of a toric variety. In other words, the number of lattice points of the polytope $P$ determines the dimension of the ambient projective space $\mathbb{P}^s$ when the (projective) toric variety is embedded into projective space.

Now we examine the most basic example of a projective toric variety, namely projective space $\mathbb{P}^n$.

Example 10 In the lattice $M = \mathbb{Z}^n$, define the standard simplex $\Delta_n \subset M_\mathbb{R}$ by

\[\Delta_n = \{(x_1, \ldots, x_n) \in \mathbb{R}^n \mid x_i \ge 0,\; x_1 + \cdots + x_n \le 1\}\]

The vertices of this polytope are $0, e_1, \ldots, e_n$, and its facets are given by the coordinate hyperplanes $\{x_i = 0\}$ and $\{x_1 + \cdots + x_n = 1\}$. The primitive inner normal vector of each facet, as an element of the dual lattice $N = \mathbb{Z}^n$, is

\[v_i = e_i \quad (i = 1, \ldots, n), \qquad v_0 = -e_1 - \cdots - e_n\]

Therefore the normal fan $\Sigma_{\Delta_n}$ has $n+1$ rays $\rho_i = \mathbb{R}_{\ge 0} v_i$ ($i = 0, 1, \ldots, n$), and its maximal cones are the $n$-dimensional cones generated by choosing $n$ of these $n+1$ rays; the case $n=2$ was illustrated in Example 2.

One could directly verify by checking the gluing that this yields $\mathbb{P}^n$, but instead let us observe the lattice points of $\Delta_n$. The lattice points of $\Delta_n$ are exactly the vertices $\{0, e_1, \ldots, e_n\}$, so the corresponding monomial map is

\[\phi_{\Delta_n}: (\mathbb{C}^\ast)^n \longrightarrow \mathbb{P}^n, \qquad (t_1, \ldots, t_n) \longmapsto [1 : t_1 : \cdots : t_n]\]

as one can verify.

Smoothness and Resolution of Singularities

Meanwhile, in §Affine Toric Varieties, ⁋Proposition 9 we saw that the smoothness of an affine toric variety $U_\sigma$ is determined by a combinatorial condition on the cone $\sigma$. Since $X_\Sigma$ is a gluing of affine charts $U_\tau$, this generalizes immediately to a criterion for general toric varieties.

Proposition 11 The toric variety $X_\Sigma$ is a smooth algebraic variety if and only if every cone $\tau \in \Sigma$ of the fan $\Sigma$ is a smooth cone.

Or more simply, it suffices to check that all maximal cones of $\Sigma$ are smooth, since the smoothness of the remaining faces follows from this. For example, in the fan of $\mathbb{P}^2$ from Example 2, the three maximal cones have determinants of the $2 \times 2$ matrices formed by adjacent pairs of rays ($\{e_1, e_2\}, \{e_2, -e_1-e_2\}, \{-e_1-e_2, e_1\}$) all equal to $\pm 1$, so $\mathbb{P}^2$ is smooth. On the other hand, consider the fan in the same lattice $\mathbb{Z}^2$ made from the three rays $(-1,-1), (2,-1), (-1,2)$; the determinants of adjacent pairs of rays are $\pm 3$, so the resulting toric variety has a $\mathbb{Z}/3$ quotient singularity in each of its three maximal cones.

A resolution of singularities for such singular toric varieties is also achieved in an explicit, combinatorial manner via refinement of the fan. First, a fan $\Sigma'$ is called a refinement of $\Sigma$ if the two fans have the same support and every cone of $\Sigma'$ is contained in some cone of $\Sigma$. In this case there is a natural toric morphism

\[\pi: X_{\Sigma'} \to X_\Sigma\]

which is a proper birational map, and if $\Sigma'$ consists entirely of smooth cones, then $\pi$ becomes a toric resolution of singularities. Such a refinement always exists for any fan, and the key points of its construction are as follows.

A non-simplicial cone can be made simplicial by star subdivision. One can think of this as appropriately dividing the cone at an interior lattice point.
For a simplicial cone $\sigma$ whose determinant is not $\pm 1$, adding a suitable lattice point as a new ray and splitting it into smaller cones reduces the determinant of each piece, and after finitely many iterations all become smooth cones.

In the above example of $\mathbb{P}^2/(\mathbb{Z}/3)$, adding the lattice points $e_1, e_2, -e_1-e_2$ located inside each of the three maximal cones as new rays causes the determinants of all maximal cones to become $\pm 1$, and the result is exactly the standard fan of $\mathbb{P}^2$. That is, $\mathbb{P}^2 \to \mathbb{P}^2/(\mathbb{Z}/3)$ is the (minimal) toric resolution of this quotient.

References

[Ful] William Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies, Princeton University Press, 1993.
[CLS] David Cox, John Little, Hal Schenck, Toric Varieties, Graduate Studies in Mathematics, American Mathematical Society, 2011.

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Definition of a Fan

Definition of a Toric Variety

The Normal Fan and Projective Toric Varieties

Smoothness and Resolution of Singularities

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