Toric intersection theory and mixed volumes

Intuition [Beginner]

Intersection theory asks how many points remain when enough subvarieties meet. On a smooth projective toric variety, the answer can be read from a polytope. A line bundle gives a lattice polytope. The top self-intersection of that line bundle is the normalised volume of the polytope. Bigger polytope, bigger degree.

For example, the line bundle $\mathcal{O}(1)$ on ${\mathbb{P}}^2$ corresponds to the unit triangle. That triangle has normalised area $1$, so a line meets a line in one point. The bundle $\mathcal{O}(d)$ corresponds to the triangle scaled by $d$. Its normalised area is $d^2$, matching the fact that two plane curves of degree $d$ meet in $d^2$ points when counted with multiplicity.

Mixed volume is the version for several different polytopes at once. If three divisors come from three polytopes, their top intersection is not the volume of one polytope. It is the mixed volume, the coefficient that measures how the volume changes when the three polytopes are added together.

Visual [Beginner]

Picture three shapes in the same lattice plane: a unit triangle $P$, a square $Q$, and their Minkowski sum $P+Q$, formed by adding every point of $P$ to every point of $Q$. As you scale $P$ and $Q$ by weights $a$ and $b$, the area of $aP+bQ$ becomes a quadratic polynomial. The coefficient of $ab$ is the mixed area. Toric intersection theory says that coefficient is the intersection number of the two corresponding divisors.

The diagram turns an algebraic operation into a geometric one. Tensoring line bundles corresponds to adding polytopes. Taking a top intersection corresponds to extracting the top-degree part of the volume polynomial.

Worked example [Beginner]

Compute the degree of the Veronese surface coming from $\mathcal{O}(2)$ on ${\mathbb{P}}^2$. The polytope for $\mathcal{O}(1)$ is the unit triangle. The polytope for $\mathcal{O}(2)$ is the same triangle scaled by $2$.

Step 1. Find the ordinary area. The unit triangle has area $1/2$. Scaling by $2$ multiplies area by $4$, so the larger triangle has area $2$.

Step 2. Normalise the area. In dimension two, normalised area is ordinary area multiplied by $2$. So the normalised area is $4$.

Step 3. Read the intersection number. The top self-intersection of $\mathcal{O}(2)$ is $4$.

What this tells us. Two general conics in the projective plane meet in $4$ points. The same number is the degree of the Veronese embedding of ${\mathbb{P}}^2$ by quadratic monomials. In toric language, both facts are the same volume calculation.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $X_{\Sigma }$ be a smooth projective toric variety of dimension $n$ with dense torus $T$, and let $D_P$ be an ample torus-invariant Cartier divisor corresponding to a full-dimensional lattice polytope $P\subset M_{\mathbb{R}}$ as in 04.11.10. The top self-intersection is $$ D_P^n=\int_{X_\Sigma} c_1(\mathcal{O}(D_P))^n. $$ The normalisation of volume is fixed by requiring a fundamental lattice simplex to have volume $1/n!$.

For convex bodies $P_1,\dots ,P_r\subset M_{\mathbb{R}}$ and non-negative real parameters ${\lambda }_1,\dots ,{\lambda }_r$, Minkowski's theorem says $$ \operatorname{Vol}(\lambda_1P_1+\cdots+\lambda_rP_r) $$ is a homogeneous polynomial of degree $n$ in the ${\lambda }_i$. The mixed volume $\mathrm{MV}(P_{i_1},\dots ,P_{i_n})$ is the symmetric multilinear coefficient determined by $$ \operatorname{Vol}(\lambda_1P_1+\cdots+\lambda_rP_r)= \sum_{i_1,\ldots,i_n}\operatorname{MV}(P_{i_1},\ldots,P_{i_n}) \lambda_{i_1}\cdots\lambda_{i_n}. $$ With this convention, $\mathrm{MV}(P,\dots ,P)=\mathrm{Vol}(P)$.

The top intersection form on ample toric divisors is the polarisation of the self-intersection polynomial: $$ D_{P_1}\cdots D_{P_n}=n!\operatorname{MV}(P_1,\ldots,P_n). $$ For a single divisor this becomes $D_P^n=n!\mathrm{Vol}(P)$.

On ${\mathbb{P}}^1\times {\mathbb{P}}^1$, the divisor $\mathcal{O}(a,b)$ corresponds to the rectangle $[0,a]\times [0,b]$. Its normalised area is $2ab$, so $$ c_1(\mathcal{O}(a,b))^2=2ab. $$ Equivalently, if $A$ and $B$ are the two ruling classes with $A^2=B^2=0$ and $AB=1$ from 04.11.12, then $(aA+bB{)}^2=2ab$.

Counterexamples to common slips

• Mixed volume is not ordinary volume of the intersection of polytopes. It is extracted from Minkowski sums, not set-theoretic intersections.

• The factor $n!$ is a normalisation, not decoration. Algebraic intersection numbers are integers for lattice polytopes. Ordinary Euclidean volume of the unit simplex is $1/n!$, so multiplying by $n!$ makes the hyperplane class on ${\mathbb{P}}^n$ have degree $1$.

• The formula uses the polytope of a divisor, not only the fan. Different ample line bundles on the same toric variety share the normal fan but have different polytopes and different intersection numbers.

Key theorem with proof [Intermediate+]

Theorem (toric mixed-volume intersection formula). Let $X_{\Sigma }$ be a smooth projective toric variety of dimension $n$, and let $D_{P_1},\dots ,D_{P_n}$ be ample torus-invariant divisors with associated lattice polytopes $P_1,\dots ,P_n$. Then $$ D_{P_1}\cdots D_{P_n}=n!\operatorname{MV}(P_1,\ldots,P_n). $$

Proof. First take the repeated case $P_1=\cdots =P_n=P$. The line bundle $\mathcal{O}(D_P)$ has global sections indexed by lattice points of $P$ 04.11.10, and the $k$-th tensor power corresponds to the dilated polytope $kP$. Thus $$ h^0(X_\Sigma,\mathcal{O}(kD_P))=|kP\cap M| $$ for all $k\ge 0$. For ample toric divisors, higher cohomology vanishes, so this is also the Hilbert polynomial for large $k$.

Ehrhart theory gives $$ |kP\cap M|=\operatorname{Vol}(P)k^n+O(k^{n-1}) $$ with lattice-normalised leading coefficient. Hirzebruch-Riemann-Roch gives the leading term of the same Hilbert polynomial as $$ \chi(X_\Sigma,\mathcal{O}(kD_P))=\frac{D_P^n}{n!}k^n+O(k^{n-1}). $$ Comparing leading coefficients gives $D_P^n=n!\mathrm{Vol}(P)$.

For distinct divisors, use polarisation. The divisor ${\lambda }_1{D}_{P_1}+\cdots +{\lambda }_r{D}_{P_r}$ corresponds, for non-negative integral ${\lambda }_i$, to the Minkowski sum ${\lambda }_1{P}_1+\cdots +{\lambda }_r{P}_r$. The repeated formula applied to this divisor gives $$ (\lambda_1D_{P_1}+\cdots+\lambda_rD_{P_r})^n =n!\operatorname{Vol}(\lambda_1P_1+\cdots+\lambda_rP_r). $$ The left side is the homogeneous intersection polynomial; the right side is the homogeneous mixed-volume polynomial. Equality for all non-negative integer ${\lambda }_i$ forces equality of coefficients. The coefficient of ${\lambda }_1\cdots {\lambda }_n$ gives the stated formula.

Bridge. The mixed-volume formula builds toward Bernstein-Kushnirenko root counting 04.11.14, where intersection points of divisors become solutions of Laurent polynomial systems, and appears again in the Duistermaat-Heckman volume computation 05.04.05. The foundational reason is that the polytope-fan dictionary identifies tensor powers with dilations, while toric cohomology 04.11.12 identifies top products with intersections. Putting these together, the bridge is the polarisation step that generalises degree from one polytope to many and identifies intersection with mixed volume.

Exercises [Intermediate+]

Advanced results [Master]

The equality $$ D_{P_1}\cdots D_{P_n}=n!\operatorname{MV}(P_1,\ldots,P_n) $$ is the polarised form of the degree formula. For a projective embedding $X_P\hookrightarrow {\mathbb{P}}^{|P\cap M|-1}$ by $L_P$, the degree of $X_P$ is $D_P^n=n!\mathrm{Vol}(P)$. The moment map of 04.11.11 gives the symplectic parallel: the pushforward of the Liouville measure is constant on $P$, so symplectic volume is the volume of the moment polytope up to the compact-torus normalisation.

The mixed-volume expression is multilinear, symmetric, translation-invariant, and monotone in each argument. These properties match algebraic intersection theory: tensor product of line bundles corresponds to Minkowski addition of polytopes; numerical equivalence ignores translation of a polytope; nef divisors have non-negative intersections. The Khovanskii-Teissier inequalities for nef divisors translate into Alexandrov-Fenchel inequalities for mixed volumes, a deep convex-geometric inequality mirrored by the Hodge index theorem.

In dimension two the mixed-area formula can be read directly from the quadratic area polynomial $$ \operatorname{Area}(aP+bQ)=a^2\operatorname{Area}(P)+2ab,\operatorname{MV}(P,Q)+b^2\operatorname{Area}(Q). $$ For toric surfaces this is the intersection form on the nef cone. On a Hirzebruch surface $F_a$, the two generators $S$ and $F$ satisfy $F^2=0$, $S\cdot F=1$, and $S^2=-a$; the polytope of $S+mF$ is a trapezoid whose area polynomial recovers exactly that intersection matrix after the support-function normalisation.

The formula is also the algebraic core of sparse elimination. A generic Laurent polynomial with Newton polytope $P_i$ cuts out a divisor $D_{P_i}$ on a toric compactification. Intersecting $n$ such divisors gives the number of isolated solutions in the dense torus after removing boundary contributions. Bernstein's theorem states that this number is $n!\mathrm{MV}(P_1,\dots ,P_n)$.

Synthesis. Toric intersection theory builds toward Bernstein-Kushnirenko 04.11.14 and appears again in tropical curve counts 04.12.05, where multiplicities are lattice volumes of local Newton polygons. The central insight is that Riemann-Roch identifies degree with the leading Hilbert coefficient, while Ehrhart theory identifies the same coefficient with volume. Putting these together identifies algebraic intersection with mixed volume and generalises the cohomology-ring product from 04.11.12.

Full proof set [Master]

Proposition (degree of a projective toric variety). Let $P\subset M_{\mathbb{R}}$ be a full-dimensional lattice polytope and let $(X_P,L_P)$ be the associated projective toric variety with ample line bundle. Then ${\mathrm{deg}}_{L_P}(X_P)=n!\mathrm{Vol}(P)$.

Proof. The degree is the top self-intersection $c_1(L_P{)}^n$. By the Demazure character formula 04.11.10, $H^0(X_P,L_P^{\otimes k})$ has basis indexed by $kP\cap M$. Hence the Hilbert function is $|kP\cap M|$ for every $k\ge 0$. Ehrhart's theorem gives leading term $\mathrm{Vol}(P)k^n$. Riemann-Roch gives leading term $c_1(L_P{)}^n{k}^n/n!$. Equality of Hilbert polynomial leading coefficients gives the claim.

Proposition (polarisation). If homogeneous degree-$n$ polynomials $F$ and $G$ agree on all non-negative integer points of ${\mathbb{Z}}^r$, then their associated symmetric multilinear forms agree.

Proof. The polynomial $F-G$ vanishes on an infinite grid in ${\mathbb{Z}}_{\ge 0}^r$. Fix all variables except one; the resulting one-variable polynomial has infinitely many zeros, so it is zero. Repeating for each variable gives $F=G$ as a polynomial. The symmetric multilinear form is recovered from the coefficients of the homogeneous polynomial by the standard polarisation identity, so the multilinear forms agree.

Proposition (mixed-volume intersection formula). For ample toric divisors $D_{P_1},\dots ,D_{P_n}$ on a common smooth projective toric variety, $D_{P_1}\cdots D_{P_n}=n!\mathrm{MV}(P_1,\dots ,P_n)$.

Proof. Apply the degree proposition to $D(\lambda )={\lambda }_1{D}_{P_1}+\cdots +{\lambda }_r{D}_{P_r}$ for all non-negative integers ${\lambda }_i$. The associated polytope is $P(\lambda )={\lambda }_1{P}_1+\cdots +{\lambda }_r{P}_r$. Therefore $$ D(\lambda)^n=n!\operatorname{Vol}(P(\lambda)). $$ The left side is the intersection polynomial in the divisor variables; the right side is $n!$ times the mixed-volume polynomial. The polarisation proposition identifies their coefficients, giving the stated mixed intersection formula.

Connections [Master]

• 04.11.12 supplies the cohomology ring in which the top products of divisor classes are computed.

• 04.11.10 supplies the polytope $P_D$ attached to a toric divisor and the lattice-point basis of global sections.

• 04.11.11 supplies the moment-polytope picture whose symplectic volume computation matches $n!\mathrm{Vol}(P)$.

• 05.04.05 gives the Duistermaat-Heckman theorem; the toric case is the symplectic shadow of this same volume formula.

• 04.12.05 uses local mixed-area determinants as tropical multiplicities, a curve-counting analogue of the mixed-volume intersection formula.

Historical & philosophical context [Master]

Kushnirenko's 1976 paper ^{[Kushnirenko 1976]} and Bernstein's 1975 paper ^{[Bernstein 1975]} identified Newton polytopes as the correct replacement for total degree in sparse polynomial systems. Danilov's 1978 survey ^{[Danilov 1978]} placed the same volume calculations inside toric intersection theory, and Fulton later made this one of the central computational payoffs of toric geometry.

Teissier's 1979 work ^{[Teissier 1979]} connected mixed volumes with intersection inequalities through the Hodge index theorem and Alexandrov-Fenchel inequalities. That line of thought explains why toric examples are not merely examples: they are a dictionary between algebraic positivity and convex-geometric positivity.

Bibliography [Master]

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}

@article{Kushnirenko1976Newton,
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}

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}

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}

@article{Teissier1979Hodge,
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}