PMATH 965: Toric Varieties

Estimated study time: 33 minutes

Table of contents

These notes treat toric varieties as a bridge between combinatorics, convex geometry, and algebraic geometry, covering fan geometry and the minimal model program together with a broader perspective on geometric invariant theory, moment maps, and equivariant cohomology. The exposition is self-contained and draws exclusively on the published literature.

Sources and References

Cox, David A., Little, John B. & Schenck, Henry K. Toric Varieties. Graduate Texts in Mathematics 124 (Springer, 2011).
Fulton, William. Introduction to Toric Varieties. Annals of Mathematics Studies 131 (Princeton University Press, 1993).
Sottile, Frank. Ibadan Lectures on Schubert Varieties (open access on the author’s Texas A&M page, math.tamu.edu/~sottile/research/html/ibadan.pdf).
Gel’fand, I. M., Zelevinsky, A. V. & Kapranov, A. V. “Newton polyhedra of principal A-determinants.” Soviet Math. Dokl. 36.2 (1988), 289–296.
Kirwan, Frances & Woolf, Jonathan. An Introduction to Intersection Homology Theory (Chapman and Hall/CRC, 1988).
Mumford, David. Geometric Invariant Theory, 3rd ed., with Fogarty and Kirwan (Springer, 1994).

Chapter 1: Toric Varieties via Monomial Maps

Toric varieties are algebraic varieties arising from a distinguished action of an algebraic torus. The most concrete entry point is through monomial maps.

1.1 The Torus and Monomial Functions

Fix an algebraically closed field \(k\) (typically \(k = \mathbb{C}\)). The algebraic torus is

\[ T = (\mathbb{G}_m)^n = (k^*)^n, \]

the group of \(n\)-tuples of nonzero scalars under coordinatewise multiplication.

A monomial is a Laurent polynomial of the form

\[ x^a = x_1^{a_1} \cdots x_n^{a_n} \]

where \(a = (a_1, \ldots, a_n) \in \mathbb{Z}^n\). The monoid of lattice points in \(\mathbb{Z}^n\) acts on \(T\) via monomial multiplication: \(a \cdot (t_1, \ldots, t_n) = (t_1^{a_1}, \ldots, t_n^{a_n})\).

Definition 1.1 (Monomial Map). Given a finite set \(A \subset \mathbb{Z}^n\), the monomial map is \[ \phi_A : T \to \mathbb{A}^m, \quad t \mapsto (t^{a_1}, \ldots, t^{a_m}), \]

where \(A = \{a_1, \ldots, a_m\}\) and \(t^{a_i} = t_1^{a_{i,1}} \cdots t_n^{a_{i,n}}\).

The closure of the image \(\phi_A(T) \subseteq \mathbb{A}^m\) is a toric variety. This is an affine toric variety. Toric varieties are obtained by gluing together such affine pieces.

1.2 The Affine Toric Variety from a Monoid

Every finitely generated submonoid \(M \subset \mathbb{Z}^n\) produces an affine toric variety. If \(M\) is generated by \(\{a_1, \ldots, a_m\}\), then

\[ X_M = \text{Spec}(k[M]), \]

where \(k[M] = k[t_1^{a_1}, \ldots, t_1^{a_m}]\) is the monoid ring.

Definition 1.2 (Monoid Ring). The monoid ring \(k[M]\) is the \(k\)-algebra with basis \(\{x^a : a \in M\}\) and multiplication given by \(x^a \cdot x^b = x^{a+b}\).

The torus \(T = (\mathbb{G}_m)^n\) acts naturally on \(X_M\) via

\[ (t_1, \ldots, t_n) \cdot x^a = (t_1^{a_1} \cdots t_n^{a_n}) x^a. \]

This action has a distinguished fixed point at the origin (when \(0 \in M\)), and the orbits correspond to faces of the monoid.

1.3 Toric Varieties from Polytopes

Toric varieties can also be constructed from lattice polytopes. A lattice polytope is a convex polytope \(P \subset \mathbb{R}^n\) with vertices in \(\mathbb{Z}^n\).

Definition 1.3 (Polytope Associated to a Monoid). Given a finitely generated monoid \(M \subset \mathbb{Z}^n\), the monoid cone is the cone \(\text{cone}(M) = \{r m : r \ge 0, m \in M\} \subset \mathbb{R}^n\). When \(M\) is the set of lattice points in a rational polytope \(P\), we recover \(P\) from the graded structure.

The polytope perspective is central to the combinatorial nature of toric geometry. Lattice points in \(P\) index global sections of the structure sheaf, and the vertices of \(P\) correspond to the toric boundary divisors.

Chapter 2: The Fan Perspective and the Orbit-Fan Correspondence

The fan, a combinatorial gadget encoding a subdivision of a cone, is the fundamental tool for understanding toric varieties globally.

2.1 Cones and Fans

A cone in \(\mathbb{R}^n\) is a set \(\sigma \subset \mathbb{R}^n\) closed under nonnegative linear combinations. A cone is rational if it can be generated by vectors in \(\mathbb{Q}^n\) (equivalently, in \(\mathbb{Z}^n\) after scaling).

Definition 2.1 (Strongly Convex Rational Polyhedral Cone). A cone \(\sigma\) is strongly convex if \(\sigma \cap (-\sigma) = \{0\}\), and rational polyhedral if it is generated by finitely many vectors in \(\mathbb{Z}^n\). The relative interior of \(\sigma\) is the interior relative to its span.

The fundamental relationship is that each cone \(\sigma\) determines an affine toric variety via the dual monoid:

\[ \sigma^\vee = \{m \in M : \langle m, v \rangle \ge 0 \text{ for all } v \in \sigma\}. \]

Here, \(M = \mathbb{Z}^n\) (or a sublattice), and \(\sigma^\vee\) is the dual cone.

Definition 2.2 (Fan). A fan in \(\mathbb{R}^n\) is a finite collection \(\Sigma\) of strongly convex rational polyhedral cones such that:

Each face of a cone in \(\Sigma\) is also in \(\Sigma\).
The intersection of any two cones in \(\Sigma\) is a face of each.

A fan \(\Sigma\) is complete if \(\bigcup_{\sigma \in \Sigma} \sigma = \mathbb{R}^n\). A fan is smooth if each cone in \(\Sigma\) can be generated by part of a \(\mathbb{Z}\)-basis of \(\mathbb{Z}^n\).

2.2 The Toric Variety from a Fan

Given a fan \(\Sigma\) in \(\mathbb{R}^n\), the toric variety \(X_\Sigma\) is constructed by gluing the affine toric varieties \(U_\sigma = \text{Spec}(k[\sigma^\vee])\) for each maximal cone \(\sigma \in \Sigma\).

Theorem 2.3 (Separation by Gluing). The toric variety \(X_\Sigma\) is a separated algebraic variety. It is quasi-compact if and only if \(\Sigma\) is complete. It is smooth if and only if each cone in \(\Sigma\) is generated by part of a \(\mathbb{Z}\)-basis.

The dimension of \(X_\Sigma\) equals the dimension of the ambient space \(\mathbb{R}^n\).

2.3 The Orbit-Fan Correspondence

The orbits of the torus action on \(X_\Sigma\) correspond bijectively to cones in the fan \(\Sigma\).

Theorem 2.4 (Orbit-Fan Correspondence). Let \(X_\Sigma\) be a toric variety with fan \(\Sigma\). There is a canonical bijection between

Torus orbits \(O_\sigma\) in \(X_\Sigma\), and
Cones \(\sigma \in \Sigma\).

The orbit \(O_\sigma\) is closed in \(X_\Sigma\) if and only if \(\sigma\) is a maximal cone. The closure \(\overline{O_\sigma}\) is isomorphic to \((\mathbb{G}_m)^{\text{rank}(\sigma^\perp \cap M)} \times \mathbb{A}^{\dim \sigma}\) where \(\sigma^\perp = \{m \in M : \langle m, \sigma \rangle = 0\}\).

This correspondence is the key to understanding toric varieties: the geometry is entirely determined by the combinatorics of the fan.

Chapter 3: Geometric Invariant Theory Perspective on Toric Varieties

The GIT perspective reveals that toric varieties are precisely the GIT quotients of the torus by finite abelian groups, or equivalently, the categorical quotients of affine space by the torus acting via monomials.

3.1 GIT Quotients and Stability

Geometric invariant theory (GIT) is a framework for constructing quotients by group actions in the category of algebraic varieties. For a reductive group \(G\) acting on an affine variety \(X = \text{Spec}(R)\), the GIT quotient is

\[ X //^{\theta} G := \text{Spec}(R^G)^{\text{ss}}, \]

where \(R^G\) is the ring of \(G\)-invariants and the superscript indicates the semistable locus.

Definition 3.1 (Toric GIT Construction). Fix a character (one-parameter subgroup) \(\lambda : \mathbb{G}_m \to T\). This induces a grading on the monoid ring \(k[M]\). The GIT quotient by \(T\) with respect to \(\lambda\) produces a toric variety; by varying \(\lambda\), one obtains different toric models of the same underlying geometry.

The toric GIT quotient approach makes explicit the dependence of the toric variety on a choice of stability condition or, more geometrically, a choice of moment map.

3.2 The Categorical Quotient

Not every quotient in classical geometry is a categorical quotient. However, for toric varieties, the quotient map is canonical.

Theorem 3.2 (Toric as Categorical Quotient). Let \(\sigma\) be a strongly convex rational polyhedral cone and \(M = \sigma^\vee \cap \mathbb{Z}^n\). The affine toric variety \(U_\sigma = \text{Spec}(k[M])\) is the categorical quotient of \(\mathbb{A}^n\) by the action of the kernel of the map \(\mathbb{Z}^n \to \mathbb{Z}^n / M\). This quotient respects the scheme structure and is functorial.

This perspective connects toric geometry to the broader theory of quotients in algebraic geometry, showing that toric varieties are among the most explicit examples of quotient singularities.

3.3 Log-Canonical Thresholds and Singularity Classes

GIT also provides a framework for understanding singularities of toric varieties. The singularities of a toric variety are entirely determined by the combinatorics of its fan.

Definition 3.3 (Gorenstein and Log-Terminal Singularities). A toric singularity is Gorenstein if the dual cone \(\sigma^\vee\) is spanned by lattice points whose convex hull contains the origin in its interior. A singularity is log-terminal if it is smooth or a quotient singularity of a smooth variety by a finite abelian group.

Toric varieties provide concrete, computable examples of these singularity classes, making them invaluable for testing invariants in higher-dimensional geometry.

Chapter 4: Torus Actions and Orbit Decomposition

The torus \(T = (\mathbb{G}_m)^n\) acts naturally on every toric variety, and this action is perhaps the single most important feature of toric geometry.

4.1 The Structure of Orbits

The torus acts on a toric variety \(X_\Sigma\) via coordinatewise multiplication. The orbits of this action are parameterized by faces of the fan, and the closure of each orbit is again a toric variety.

Theorem 4.1 (Orbit Closure Structure). For a cone \(\sigma \in \Sigma\), the orbit \(O_\sigma\) is isomorphic to a torus \((\mathbb{G}_m)^{\text{rank}(M / \sigma^\perp)}\) where \(\sigma^\perp = \{m \in M : \langle m, v \rangle = 0 \text{ for all } v \in \sigma\}\). The closure \(\overline{O_\sigma}\) is a closed toric subvariety isomorphic to \(X_{\Sigma_\sigma}\) where \(\Sigma_\sigma\) is the link of \(\sigma\).

Orbit closures are smooth if and only if the corresponding cone is generated by part of a lattice basis.

4.2 Torus-Invariant Subvarieties

Every closed torus-invariant subvariety of a toric variety corresponds to a face of the fan.

Proposition 4.2 (Torus-Invariant Subvarieties and Faces). The torus-invariant closed subvarieties of \(X_\Sigma\) are in bijection with cones in the fan \(\Sigma\). If \(\sigma \in \Sigma\), the corresponding subvariety is \(\overline{O_\sigma} \cong X_{\Sigma_\sigma}\).

This allows one to study the structure of toric varieties entirely through the combinatorics of the fan.

4.3 Fixed Points and Moment Graphs

When a one-parameter subgroup \(\lambda : \mathbb{G}_m \to T\) acts on \(X_\Sigma\), the fixed-point locus is a union of orbit closures corresponding to cones \(\sigma\) such that \(\lambda(\mathbb{G}_m) \cap (\sigma^\perp \otimes \mathbb{R})\) spans \(\sigma^\perp\).

The moment graph is a combinatorial graph encoding the fixed-point structure and the weight decomposition of the normal bundle at fixed points. This is useful for understanding the action and for computing equivariant cohomology.

Chapter 5: Moment Maps and Symplectic Aspects

Toric varieties admit a natural symplectic structure that respects the torus action, enabling the use of moment map techniques from symplectic geometry.

5.1 The Moment Map

When \(k = \mathbb{C}\), a toric variety \(X_\Sigma\) admits a \(\mathbb{R}_{>0}^n\)-action, and the associated moment map is a crucial invariant.

Definition 5.1 (Moment Map for Toric Varieties). The moment map is a continuous map \(\mu : X_\Sigma(\mathbb{C}) \to \mathbb{R}^n\) defined on the topological space of complex points. The image is the union of the cones \(\sigma \in \Sigma\), viewed as subsets of \(\mathbb{R}^n\).

More precisely, if \(\lambda : \mathbb{G}_m \to T\) is a one-parameter subgroup corresponding to \(u \in \mathbb{Z}^n\), the associated moment function is linear on each orbit closure.

5.2 Kähler Geometry and the Fano Case

For Fano toric varieties (those corresponding to complete fans where all cones are strictly convex), the variety admits a \(\mathbb{G}_m\)-invariant Kähler-Einstein metric.

Definition 5.2 (Toric Kähler-Einstein Metrics). A toric Fano variety \(X_\Sigma\) admits a \(\mathbb{G}_m\)-invariant Kähler-Einstein metric (a Kähler metric with Ricci curvature proportional to the metric form) if and only if the fan \(\Sigma\) satisfies certain combinatorial balancing conditions.

This connects to the Yau-Tian-Donaldson conjecture and provides explicit examples where Kähler-Einstein metrics exist on singular varieties.

5.3 Symplectic Reduction and Convex Polytopes

The symplectic viewpoint reveals that toric varieties arising from complete fans correspond to symplectic reductions of \(\mathbb{C}^m\) by a compact torus action.

Theorem 5.3 (Symplectic Cut and Polytope Correspondence). A complete toric variety \(X_\Sigma\) of dimension \(n\) is symplectomorphic to the symplectic reduction of \(\mathbb{C}^m\) (where \(m = |\Sigma^{(n-1)}|\) is the number of \((n-1)\)-dimensional cones, i.e., edges of the fan) by a lattice quotient of \((\mathbb{G}_m)^m\). The moment map image is the Newton polytope associated to \(\Sigma\).

Chapter 6: Divisors—Weil, Cartier, and Torus-Equivariant

Divisors are the primary tool for understanding the geometry of toric varieties. In the toric setting, divisor theory becomes entirely combinatorial.

6.1 Torus-Invariant Prime Divisors

The torus-invariant prime divisors correspond to the \((n-1)\)-dimensional cones in the fan.

Definition 6.1 (Toric Divisors). Let \(\Sigma^{(n-1)}\) denote the set of \((n-1)\)-dimensional cones (the facets) of the fan. For each \(\tau \in \Sigma^{(n-1)}\), there is a corresponding torus-invariant prime divisor \(D_\tau\). Every torus-invariant divisor is a \(\mathbb{Z}\)-linear combination \(\sum_{\tau \in \Sigma^{(n-1))} a_\tau D_\tau\).

The closure of each orbit of codimension 1 is a divisor, and every torus-invariant divisor is a union of these prime divisors.

6.2 Weil and Cartier Divisors

In toric varieties, Weil and Cartier divisors coincide precisely when the variety is normal, which is always the case for toric varieties.

Theorem 6.2 (Divisor Coincidence in Toric Varieties). For a toric variety \(X_\Sigma\), every Weil divisor is Cartier, and the group of Weil divisors is isomorphic to the free abelian group generated by the torus-invariant prime divisors.

This is because toric varieties are normal, and on a normal variety, every Weil divisor is Cartier if and only if the variety is regular in codimension 1.

6.3 Torus Action on Line Bundles and Polytopes

A torus-invariant Cartier divisor on \(X_\Sigma\) is determined by a piecewise linear function on the fan, which is dual to a polytope.

Definition 6.3 (Polytope Associated to a Divisor). A torus-invariant divisor \(D = \sum_{\tau \in \Sigma^{(n-1)}} a_\tau D_\tau\) determines a polytope in \(\mathbb{R}^n\) by collecting all supporting hyperplanes \(\{m \in M \otimes \mathbb{R} : \langle m, u_\tau \rangle \ge a_\tau\}\) where \(u_\tau\) is the primitive inward normal to the facet \(\tau\).

The global sections of the line bundle \(\mathcal{O}(D)\) correspond to lattice points in this polytope, giving a direct combinatorial description of the space of sections.

Chapter 7: Line Bundles and Ampleness Conditions

Ample line bundles on toric varieties are characterized combinatorially in terms of the associated polytopes.

7.1 The Ample Cone

A divisor \(D\) on a projective toric variety is ample if its associated polytope \(P_D\) contains a strictly interior lattice point.

Definition 7.1 (Ampleness for Toric Divisors). A torus-invariant divisor \(D\) on a complete toric variety \(X_\Sigma\) is ample if:

The polytope \(P_D = \{m \in M \otimes \mathbb{R} : \langle m, u_\tau \rangle \ge a_\tau \text{ for all } \tau\}\) contains a strictly interior lattice point, and
For each facet of the fan, the hyperplane supporting \(P_D\) meets the facet transversely.

The ample cone is a rational polyhedral cone in the vector space of torus-invariant divisors, and it is the interior of the effective cone.

7.2 Very Ample and Embedding Properties

A line bundle \(\mathcal{L}\) is very ample if the sections of \(\mathcal{L}\) give an embedding into projective space. For toric varieties, this is characterized by properties of the polytope.

Theorem 7.2 (Very Ample for Toric Varieties). A torus-invariant ample divisor \(D\) on a complete toric variety \(X_\Sigma\) is very ample if and only if the associated polytope \(P_D\) is lattice polytope (i.e., all vertices are lattice points) and, for each facet \(\tau\) of the fan, the supporting hyperplane of \(P_D\) is a lattice hyperplane with respect to the lattice in that facet.

This allows one to detect when toric varieties embed into projective space by purely combinatorial means.

7.3 The Picard Group of a Toric Variety

The Picard group of a toric variety is the quotient of the divisor group by principal divisors, and it has a particularly nice structure.

Theorem 7.3 (Picard Group Structure). For a toric variety \(X_\Sigma\), the Picard group \(\text{Pic}(X_\Sigma)\) is isomorphic to the quotient \(M^\vee / N\) where \(M^\vee\) is the dual lattice of characters and \(N\) is the subgroup of characters that are constant on the fan. When \(\Sigma\) is complete, \(\text{Pic}(X_\Sigma) \cong \mathbb{Z}^{\#(\Sigma^{(n-1)})}/\text{(relations from the fan structure)}\).

Chapter 8: Cohomology of Toric Varieties

The cohomology of toric varieties is computed using both classical methods and equivariant techniques.

8.1 Singular Cohomology

The topological structure of a complete toric variety is determined by the combinatorics of its fan. The singularities visible in cohomology are reflected in the fan’s structure.

Theorem 8.1 (Dolbeault Cohomology and Hodge Diamond). For a smooth complete toric variety \(X_\Sigma\) of dimension \(n\), the Hodge numbers \(h^{p,q} = \dim H^{p,q}(X_\Sigma)\) satisfy:

\(h^{p,q} = 0\) unless \(0 \le p, q \le n\).
\(h^{p,q} = h^{n-p, n-q}\) (Serre duality).
For the toric case, \(h^{p,q}\) depends only on the combinatorics of \(\Sigma\).

8.2 Equivariant Cohomology

The torus action on a toric variety gives rise to equivariant cohomology with respect to the torus. This is computed via the equivariant Dolbeault complex or via the Atiyah-Segal isomorphism.

Definition 8.2 (Equivariant Cohomology of Toric Varieties). The T-equivariant cohomology of a toric variety \(X_\Sigma\) is the cohomology of the complex \(\Omega^*_{X_\Sigma} \otimes S(\mathfrak{t}^*)\) where \(\mathfrak{t}^*\) is the character group of \(T\). The result is a module over the symmetric algebra \(S(\mathfrak{t}^*)\), and it contains all the classical cohomology information.

Equivariant cohomology is particularly powerful for toric varieties because it can be computed combinatorially using the moment graph.

8.3 GKM Theory and the Moment Graph

The Goresky-Kottwitz-MacPherson (GKM) theory provides an algorithm for computing equivariant cohomology from the fixed-point set and the equivariant Poincaré pairing.

Theorem 8.3 (GKM Theory for Toric Varieties). For a smooth complete toric variety \(X_\Sigma\), the equivariant cohomology \(H^*_T(X_\Sigma)\) is isomorphic to the subalgebra of \(\prod_{\sigma \in \Sigma^{(0)}} H^*_T(pt)\) consisting of sequences that are compatible with the edge structure of the moment graph (the graph of edges in the fan, with weights determined by one-parameter subgroups).

Chapter 9: Toric Morphisms and Resolutions of Singularities

The morphisms between toric varieties correspond to maps between fans satisfying a compatibility condition. Resolutions of singularities are obtained by subdividing the fan.

A morphism of toric varieties \(X_{\Sigma'} \to X_\Sigma\) corresponds to a map of fans, which is a subdivision map satisfying a linearity condition.

Definition 9.1 (Toric Morphism and Fan Refinement). A morphism of toric varieties \(\phi: X_{\Sigma'} \to X_\Sigma\) is toric if it commutes with the torus actions. This corresponds to a refinement of the fan: a map \(\Sigma' \to \Sigma\) such that for each cone \(\sigma' \in \Sigma'\), there exists \(\sigma \in \Sigma\) with \(\sigma' \subseteq \sigma\) (as sets in \(\mathbb{R}^n\)).

Every refinement of a fan produces a toric morphism. The morphism is proper if and only if the refinement is complete-to-complete.

9.2 Resolving Singularities by Fan Subdivision

The primary method for resolving singularities in toric geometry is to subdivide the fan, replacing each singular cone with smaller smooth cones.

Theorem 9.2 (Existence of Smooth Toric Resolution). Every normal toric variety admits a smooth toric resolution of singularities obtained by subdividing the fan into smooth cones. The resulting morphism is an isomorphism over the smooth locus of the original variety.

The resolution is not always minimal; finding minimal resolutions requires understanding which subdivisions remove the fewest singularities necessary.

9.3 Orbifold Resolutions and Quotient Singularities

For toric varieties with quotient singularities (abelian quotient singularities), the minimal resolution is obtained by a canonical fan subdivision related to the quotient structure.

Proposition 9.3 (Minimal Resolution of Quotient Singularities). If \(X_\sigma\) is a quotient singularity arising from a cone \(\sigma\), the minimal resolution is obtained by subdividing \(\sigma\) using the lattice points on the edges of the cone's primitive generators. The exceptional divisors are rational curves meeting transversally according to the Dynkin diagram structure of the quotient group.

Chapter 10: The Minimal Model Program for Toric Varieties

The minimal model program (MMP) is a program in birational geometry aimed at reducing varieties to canonical or minimal models. Toric varieties provide explicit examples where the MMP can be studied in detail.

10.1 Birational Transformations and the Cone of Curves

The cone of effective curves (or Mori cone) is the cone generated by numerical classes of effective curves. The cone of ample divisors is dual to the cone of effective curves.

Definition 10.1 (Mori Cone of a Toric Variety). For a complete normal toric variety \(X_\Sigma\), the Mori cone \(\overline{NE}(X_\Sigma)\) is generated by the rays \(\mathbb{R}_{\ge 0}[C_\tau]\) where \(C_\tau\) are the toric prime divisors (corresponding to \((n-1)\)-dimensional cones \(\tau\) of the fan). The cone of ample divisors is the dual cone.

The structure of this cone determines the birational geometry of the variety.

10.2 Contraction of Faces and K-Trivial Varieties

A central theme in the MMP is the contraction of extremal faces of the Mori cone. For a K-trivial (Calabi-Yau) toric variety, these contractions are particularly well-behaved.

Theorem 10.2 (MMP for Toric Varieties). A complete toric variety admits a minimal model program: a sequence of divisorial contractions or flips that produce either a Mori dream space or a variety with nef canonical divisor. In the toric setting, these operations correspond to explicit subdivisions and refinements of the fan.

Toric Calabi-Yau varieties (K-trivial complete toric varieties) are particularly important in mirror symmetry and string theory.

10.3 The Secondary Fan and Flip Structure

The secondary fan is a fan in the space of divisor classes whose cones correspond to different toric models of the same underlying abstract variety.

Definition 10.3 (Secondary Fan and Wall Crossing). Given a fixed lattice \(M\) and a fixed polytope \(P\) in \(M \otimes \mathbb{R}\), the secondary fan is a fan in the space \(M\) whose cones correspond to regular subdivisions of \(P\). Moving from one chamber of the secondary fan to another corresponds to a biregular operation (a "flip" in the sense of the MMP).

The secondary fan encodes all possible toric models of a given combinatorial type and is crucial for understanding the structure of Schubert varieties and other classical varieties.

Chapter 11: Applications—Toric Orbifolds, Polytopes, Secondary Fans, and Schubert Varieties

Toric varieties provide a testing ground and a bridge between combinatorics and classical algebraic geometry.

11.1 Toric Orbifolds and Stacky Resolutions

When the fan is not smooth, the associated toric variety has singularities. One can also work with toric stacks, which assign a finite group quotient to each cone.

Definition 11.1 (Toric Orbifold as Orbifold Stack). A toric orbifold is a toric stack, parameterized by a fan in \(\mathbb{R}^n\) where each maximal cone is allowed to have a nontrivial stabilizer subgroup (a finite subgroup of \(GL_n(\mathbb{Z})\)). The coarse moduli space is a toric variety with quotient singularities, and the stack structure encodes the orbifold singularities.

Toric orbifolds arise naturally in string theory and enumerative geometry, where one counts curves in quotient singularities.

11.2 Lattice Polytopes and Integer Points

The polytope associated to a toric variety encodes much of its geometry. The number of lattice points in the polytope is related to the dimension of spaces of global sections.

Theorem 11.2 (Ehrhart Polynomial and Sections). For a lattice polytope \(P \subset M \otimes \mathbb{R}\), the Ehrhart polynomial is the polynomial \(E_P(t) = \#(tP \cap M)\) for \(t \in \mathbb{Z}_{\ge 0}\). For the associated toric variety and its ample divisor, the dimension of global sections is given by \(E_P(t) - 1\) (minus 1 for the projective normalization), and the volume of \(P\) equals the arithmetic genus.

This connects combinatorial properties (lattice point counts, volumes) directly to algebraic-geometric invariants (dimensions of cohomology, degrees).

11.3 Secondary Fans and Flip Chambers

The secondary fan of a lattice polytope parameterizes all regular subdivisions, and moving from one chamber to another corresponds to a geometric transition (a “flip” of the polytope).

Definition 11.3 (Regular Subdivision and Secondary Fan Chamber). A regular subdivision of a lattice polytope \(P\) is a polyhedral subdivision induced by the lower faces of the convex hull of lifted points (the height function). The secondary fan's cones correspond to combinatorially distinct regular subdivisions, and its structure encodes how these subdivisions relate via flips.

The secondary fan is useful for understanding the Gröbner fan (which encodes toric initial ideals) and for enumerative questions about triangulations of polytopes.

11.4 Toric Limits and Schubert Varieties

A classical Schubert variety (the closure of a Borel orbit in the flag variety \(GL_n/B\)) is not toric in general. However, one can construct a toric limit or toric degeneration of a Schubert variety, which is a toric variety that captures some of the combinatorial structure of the Schubert variety.

Definition 11.4 (Toric Degeneration). A toric degeneration of a variety \(X\) is a flat family \(\mathcal{X} \to \mathbb{A}^1\) such that the generic fiber is isomorphic to \(X\) and the special fiber is a toric variety \(X_{\text{toric}}\). The toric variety encodes combinatorial information about \(X\) (such as the moment polytope or the structure of vanishing cycles).

For Schubert varieties, toric degenerations are given by the Gröbner basis construction, where one uses a monomial order to degenerate the ideal to a toric ideal. The resulting toric variety is the toric ideal Gröbner degeneration, whose combinatorics (encoded in the secondary fan of the moment polytope) mirrors the combinatorics of the Schubert variety (the poset of Schubert cells).

Toric degenerations are powerful tools for studying the cohomology and intersections on classical varieties through toric geometry.