Overview
This lecture introduces toric geometry, focusing on the combinatorial objects (lattices, cones, polytopes, fans) underlying toric varieties and their applications in polynomial equation solving and algebraic geometry.
Introduction to Toric Varieties
- Toric varieties are algebraic varieties encoded by combinatorial data like lattices, cones, polytopes, and fans.
- Many familiar varieties (e.g., affine and projective spaces, Hirzebruch surfaces) are toric.
- Toric geometry connects algebraic geometry, polyhedral geometry, tropical geometry, and invariant theory.
- Solutions to polynomial systems with fixed monomial support naturally live in toric varieties.
Affine Toric Varieties
- An affine toric variety is an irreducible affine variety containing a dense torus (C*)ⁿ with an algebraic torus action.
- Three equivalent constructions: via monomial maps, toric ideals (binomial ideals), or affine semigroups (semigroup algebras).
- Every affine toric variety can be expressed as Specm(C[S]) for an affine semigroup S.
- Nicest affine toric varieties correspond to semigroups from rational convex polyhedral cones; normality and smoothness relate to properties of these cones.
Lattices, Tori, and Cones
- The character lattice M ≅ ℤⁿ consists of homomorphisms from the torus to C*; its dual, N, is the cocharacter lattice.
- Rational convex polyhedral cones are generated by finite subsets of N and have duals and faces.
- Strongly convex cones (pointed) give rise to affine toric varieties with unique torus-fixed points.
Projective Toric Varieties and Polytopes
- Projective toric varieties are closures of monomial maps into projective space, containing a dense torus.
- Each projective toric variety corresponds combinatorially to a convex lattice polytope.
- Polytope properties (dimension, degree, normality, very ampleness) govern geometric features of the variety.
Kushnirenko’s Theorem & Polynomial Systems
- The degree of a projective toric variety equals the normalized volume n!Vol(P) of its associated polytope P.
- The number of solutions to a system of Laurent polynomials with exponents in A is bounded by n!Vol(P).
Abstract Toric Varieties from Fans
- Fans (collections of strongly convex rational cones) encode how to glue affine toric varieties to produce general toric varieties.
- The orbit-cone correspondence links cones in the fan to torus orbits in the variety, stratifying the space.
Divisors on Toric Varieties
- Weil divisors on toric varieties correspond to torus-invariant prime divisors indexed by rays of the fan.
- The class group Cl(X) and Picard group Pic(X) have explicit combinatorial descriptions in terms of the fan and lattice.
- Cartier divisors correspond to certain combinatorial data or global sections over the fan.
Homogeneous Coordinates & Cox Rings
- Toric varieties admit homogeneous coordinates via the Cox ring (homogeneous coordinate ring), graded by the class group.
- Toric varieties can be described as certain GIT (geometric invariant theory) quotients of affine spaces minus “irrelevant loci” by reductive groups.
Polynomial Systems & BKK Theorem
- Bernstein-Kushnirenko-Khovanskii (BKK) Theorem generalizes solution counts for polynomial systems to “mixed volumes” of associated polytopes.
- Minkowski sums and mixed volumes determine the number of solutions for generic systems with multiple supports.
Key Terms & Definitions
- Torus — (C*)ⁿ with group structure and dense open embedding in the variety.
- Character lattice (M) — the group of monomial maps T → C*, isomorphic to ℤⁿ.
- Affine toric variety — an affine variety with a dense torus and torus action extending to the whole variety.
- Toric ideal — a prime ideal in C[x₁, …, xₛ] generated by binomials.
- Affine semigroup — finitely generated, commutative, embedded in a lattice; basis for affine toric varieties.
- Rational convex polyhedral cone — a cone in Nℝ generated by lattice vectors.
- Fan — finite collection of cones fitting together nicely, encoding gluing data for toric varieties.
- Weil divisor — formal sum of codimension-one subvarieties.
- Cartier divisor — locally principal Weil divisor; can be described by global sections.
- Class group / Picard group — groups of divisors modulo linear (resp. Cartier) equivalence.
- Cox ring — homogeneous coordinate ring for a toric variety, graded by the class group.
- Minkowski sum — sum of polytopes: {p + q | p ∈ P, q ∈ Q}.
- Mixed volume — coefficient in the polynomial expansion of volumes of Minkowski sums; counts generic solutions.
Action Items / Next Steps
- Review sections 2–6 for deeper computational examples and to practice with Oscar.jl.
- Attempt the exercises for hands-on understanding of the constructions.
- Explore provided MathRepo and Julia notebooks for computational exploration.