Many geometric properties of a toric variety are equivalent to explicit conditions on its fan.
Smooth, complete, projective, Fano, read off the fan (Tag PR-FULCRIT)
Let \(X_\Sigma\) be the toric variety of a fan \(\Sigma\) in \(N_\RR \cong \RR^n\).
| \(X_\Sigma\) has the property | exactly when \(\Sigma\) satisfies |
|---|---|
| normal | always: each \(S_\sigma\) is saturated |
| smooth | every cone is spanned by part of a \(\ZZ\)-basis of \(N\); in dimension \(n\), \(\det = \pm 1\) |
| simplicial, equivalently \(\QQ\)-factorial, equivalently finite quotient singularities only | every cone is spanned by \(\RR\)-independent vectors, that is \(\det \neq 0\) |
| complete, equivalently compact | \(\abs{\Sigma} = N_\RR\) |
| projective | \(\Sigma\) is the normal fan of a lattice polytope, equivalently some \(D\) has strictly convex support function |
| Fano | \(\Sigma\) is the fan of cones over the faces of a reflexive polytope |
For a divisor \(D = \sum a_\rho D_\rho\) with support function \(\varphi_D\) on a complete \(\Sigma\):
| \(D\) has the property | exactly when |
|---|---|
| base point free | \(\varphi_D\) is convex |
| ample | \(\varphi_D\) is strictly convex |
| very ample | ample, and each vertex semigroup \(\ts{m - m_\sigma \st m \in P_D \intersect M}\) is saturated in \(M\) |
(How the rows depend on each other)
Smooth implies simplicial, so a cone that is not simplicial is singular before any determinant is computed. Projective implies complete, and the converse fails: a complete fan admitting no strictly convex support function gives a proper variety that is not projective. Fano implies projective, since \(-K_X\) ample is itself a strictly convex support function.
Two rows collapse in low dimension. On a smooth complete toric variety ample and very ample agree, and on any complete toric surface they agree as well, so a counterexample needs dimension three and a singular point.
(Worked positivity on two families)
On \(\PP^n\) with \(D = \sum a_i D_i\), base point free means \(\sum a_i \geq 0\) and ample means \(\sum a_i > 0\).
On \(\FF_m\) with rays labelled as \(D_1, \ldots, D_4\) in cyclic order, \(\Pic(\FF_m) = \gens{D_1, D_4}\), and \(D = a D_1 + b D_4\) is ample exactly when \(a > 0\) and \(b > 0\).
\(X_\Sigma\) is complete exactly when \(\abs\Sigma=N_\RR\), projective exactly when \(\Sigma\) is the normal fan of a full-dimensional lattice polytope, and Gorenstein Fano exactly when \(\Sigma\) is the face fan of a reflexive polytope.
Betti numbers and degree
Betti numbers of a smooth complete toric variety from the cone counts (Tag T-FULBETTI)
Let \(X_\Sigma\) be smooth and complete of dimension \(n\), and write \(\Sigma(k)\) for the set of \(k\)-dimensional cones. Then the odd Betti numbers vanish and \[ \beta_{2k} = \sum_{i=k}^{n} (-1)^{i-k} \binom{i}{k} \size \Sigma(n-i) . \]
(Two special cases)
\[ \beta_0 = \beta_{2n} = 1, \qquad \chi(X_\Sigma) = \size \Sigma(n) , \] and when every maximal cone has dimension \(n\), \[ H^2(X_\Sigma; \ZZ) \cong \Pic(X_\Sigma), \qquad \beta_2 = \size \Sigma(1) - n . \]
(Where the formula comes from)
Cohomology of an affine piece is exterior algebra on the lattice orthogonal to the cone, \[ H^i(U_\sigma; \ZZ) \cong \Extpower^i M(\sigma), \qquad M(\sigma) \da \sigma^\perp \intersect M , \] so the Mayer–Vietoris spectral sequence for the cover by the \(U_\sigma\) over maximal cones has \[ E_1^{p,q} = \bigoplus_{i_0 < \cdots < i_p} \Extpower^q M(\sigma_{i_0} \intersect \cdots \intersect \sigma_{i_p}) \abuts H^{p+q}(X_\Sigma; \ZZ) . \] The alternating sum over \(q\) of \(\rank \Extpower^q M(\tau)\) is \(0\) unless \(\dim \tau = n\), where it is \(1\), and only the maximal cones survive. That is the Euler characteristic statement, and tracking the individual ranks gives the binomial formula.
(Check it on the two standard surfaces)
For \(\PP^2\): \(\size \Sigma(2) = 3\), \(\size \Sigma(1) = 3\), \(\size \Sigma(0) = 1\), giving \(\beta_0 = 1\), \(\beta_2 = 1\), \(\beta_4 = 1\) and \(\chi = 3\). For \(\FF_a\): \(\size \Sigma(2) = 4\), \(\size \Sigma(1) = 4\), giving \(\beta_2 = 2 = \rank \Pic(\FF_a)\) and \(\chi = 4\).
The formula turns cohomology into cone counts, while the torus-orbit cell decomposition explains structurally why the odd cohomology vanishes.
The degree of a projective toric variety is the normalised volume of its polytope (Tag T-FULDEG)
Let \(P \subseteq M_\RR\) be a full-dimensional very ample lattice polytope of dimension \(n\), embedded by the lattice points of \(P\). Then \[ \deg X_P = n! \cdot \vol(P) , \] the volume normalised so that the standard simplex has volume \(1/n!\). Equivalently \(\deg X_P = D_P^{\,n}\), the top self-intersection of the ample divisor.
(Ehrhart, in one line)
The statement is the leading term of the lattice-point count. Since \(h^0(X_P, \OO(kD_P)) = \size(kP \intersect M)\) and Ehrhart’s theorem makes this a polynomial in \(k\) of degree \(n\) with leading coefficient \(\vol(P)\), comparison with the Hilbert polynomial of \(X_P\) gives the degree. So the same lattice-point count that computes sections also computes the degree; only the leading coefficient is read instead of the value at \(k = 1\).
(Basic checks)
For \(P = \Delta\), the standard simplex, \(\vol = 1/n!\) and \(\deg \PP^n = 1\). For \(P = d\Delta\), \(\vol = d^n/n!\) and \(\deg = d^n\), which is the degree of the \(d\)-uple Veronese image of \(\PP^n\). For the unit square, \(2! \cdot 1 = 2\), the degree of the quadric surface \(\PP^1 \times \PP^1 \subseteq \PP^3\).
For \(X_\Sigma\) smooth and complete of dimension \(n\), with \(d_j\) the number of \(j\)-dimensional cones of \(\Sigma\), \(b_{2k}=\sum_{i=k}^n(-1)^{i-k}\binom ik d_{n-i}\) and the odd Betti numbers vanish. For an ample torus-invariant divisor \(D\) with polytope \(P_D\), \(h^0(\OO(D))=\#(P_D\cap M)\) and \(D^n=n!\operatorname{vol}(P_D)\).
Cohen–Macaulay and Gorenstein toric varieties
Toric varieties are rational and Cohen-Macaulay, and affine ones have no nontrivial vector bundles (Tag PR-FULCM)
Let \(X_\Sigma\) be a toric variety of dimension \(n\).
\(X_\Sigma\) is rational: it contains the torus \(T \cong \GG_m^n\) as a dense open set, so \(k(X_\Sigma) = k(t_1, \ldots, t_n)\).
\(X_\Sigma\) is Cohen-Macaulay: every local ring \(R\) has \(\depth R = \dim R\).
Every vector bundle on an affine toric variety \(U_\sigma\) is trivial, equivalently every finitely generated projective \(k[S_\sigma]\)-module is free.
(Consequences)
Rationality implies that toric varieties cannot furnish unirational-but-nonrational examples: every toric variety is already rational.
Cohen-Macaulayness is the input that makes Serre duality and the vanishing theorems available without extra hypotheses, and it is why \(\omega_{X_\Sigma} = \OO(-\sum_\rho D_\rho)\) behaves as a dualising sheaf even at singular points. Note the contrast with Gorenstein, which is a genuine condition: \(X_\Sigma\) is Gorenstein exactly when \(K_X\) is Cartier, which for \(X_P\) means \(P\) reflexive.
The third statement is the toric case of a theorem on semigroup rings, and it says that \(\Pic(U_\sigma) = 0\) for every cone, which is what makes the class group of the cone over the rational normal curve a clean example of \(\Pic \subsetneq \Cl\).
Every normal toric variety is Cohen–Macaulay. \(U_\sigma\) is Gorenstein exactly when some \(m\in M\) has \(\langle m,u_\rho\rangle=1\) for every minimal ray generator \(u_\rho\) of \(\sigma\).