Comments on toric varieties

Here are few notes on not necessarily normal toric varieties and resolution by toric blow-up. These notes are independent of, but in the same spirit as the earlier preprint [Tho03]. That is, they focus on the fact that toric varieties are locally given by monoid algebras. 1 Not necessarily normal toric varieties Fix a base field k. We start by recalling the definition of a fan. Let N ∼= Z be a lattice, that is, a finitely generated free Abelian group. A (strongly convex rational polyhedral) cone, σ, in NR = N ⊗ Z R is a set consisting of all nonnegative linear combinations of some fixed finite set of vectors in the lattice, σ = R≥0v1 + · · ·R≥0vr, v1, . . . , vr ∈ N that contains no line. Here we identify N with its image, {n⊗1 | n ∈ N}, in NR. Let M = Hom(N,Z) be the dual lattice to N , identify M with its image in MR, identify MR with the dual space to NR, and let 〈, 〉 be the dual pairing. We say a d − 1-dimensional subspace, H, of NR is a supporting hyperplane of σ if there exists a vector u ∈ MR such that H = {v | 〈u, v〉 = 0} and σ ⊂ {v | 〈u, v〉 ≥ 0}. A face of a cone σ is a subset of the form H ∩ σ where H is a supporting hyperplane of σ. A fan, ∆ is a finite collection of cones that is closed under taking faces such that the intersection of any two cones in σ is a face of each. To each cone σ, we associate: (1) a finitely generated submonoid Sσ = σ ∩M of M , where σ = {u ∈ MR | 〈u, v〉 ≥ 0, ∀v ∈ σ}; (2) the finitely generated k-algebra k[Sσ]; and, (3) the affine k-variety Uσ = Spec k[Sσ]. The (affine) toric variety associated to σ is Uσ. If τ is a face of σ, k[Sτ ] is a localization of k[Sσ] and Uτ is an open affine subset of Uσ. Using