Chapter 1 Affine Toric Varieties §1.0. Background: Affine Varieties We begin with the algebraic geometry needed for our study of affine toric varieties. Our discussion assumes Chapters 1–5 and 9 of [69]. Coordinate Rings. An ideal I ⊆ S = C[x1,...,xn] gives an affine variety V(I) = {p ∈ Cn | f (p) = 0 for all f ∈ I} and an affine variety V ⊆ Cn gives the ideal I(V) = { f ∈ S | f (p) = 0 for all p ∈ V }. By the Hilbert basis theorem, an affine variety V is defined by the vanishing of finitely many polynomials in S, and for any ideal I, the Nullstellensatz tells us that I(V(I)) = √ I = { f ∈ S | f ∈ I for some ≥ 1} since C is algebraically closed. The most important algebraic object associated to V is its coordinate ring C[V] = S/I(V). Elements of C[V] can be interpreted as the C-valued polynomial functions on V. Note that C[V] is a C-algebra, meaning that its vector space structure is compatible with its ring structure. Here are some basic facts about coordinate rings: • C[V] is an integral domain ⇔ I(V) is a prime ideal ⇔ V is irreducible. • Polynomial maps (also called morphisms) φ : V1 → V2 between affine varieties correspond to C-algebra homomorphisms φ∗ : C[V2] → C[V1], where φ∗(g) = g ◦ φ for g ∈ C[V2]. • Two affine varieties are isomorphic if and only if their coordinate rings are isomorphic C-algebras. 3

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