Week 2: Local Properties of Toric Varieties (DRAFT)

  1. Distinguished points in XσX_\sigma

These notes correspond to the second week of the toric geometry learning seminar, presented by Abhishek Koparde.

Distinguished points in XσX_\sigma

Let's start with a lemma.

Lemma. There exists a bijective correspondence

{ closed points of Xσ }↔{ semi group homomorphisms Sσ→C }.\big\{~\text{closed points of } X_\sigma~\big\} \leftrightarrow \big\{~\text{semi group homomorphisms } S_\sigma \to \mathbb{C}~\big\}.

Proof: By definition, each closed point of XσX_\sigma is a morphism φ:Spec⁡C→Spec⁡C[Sσ]\varphi:\operatorname{Spec}\mathbb{C} \to \operatorname{Spec} \mathbb{C}[S_\sigma] and each of these corresponds to a map of rings φ∗:Spec⁡C[Sσ]→C\varphi_*:\operatorname{Spec}\mathbb{C}[S_\sigma] \to \mathbb{C}. Given such a ring map, we obtain a semigroup homomorphism ψ:Sσ→C\psi:S_\sigma \to \mathbb{C} by sending ψ(s)=φ(xs)\psi(s) = \varphi(x^s).

Conversely, given a semigroup homomorphism ψ:Sσ→C\psi:S_\sigma \to \mathbb{C}, we obtain a ring homomorphism

φ∗:C[Sσ]→C\varphi_*:\mathbb{C}[S_\sigma] \to \mathbb{C}

by defining φ∗(xs)=ψ(s)\varphi_*(x^s) = \psi(s) for each s∈Sσs \in S_\sigma and extending linearly. Note that in each case we take C\mathbb{C} to be a semigroup with respect to multiplication, NOT addition. □\hspace{19.5em}\square

For each cone in a toric variety there is a particular closed point in which we'll be interested. Consider the semigroup morphism φσ:Sσ→C\varphi_\sigma: S_\sigma \to \mathbb{C} defined

φσ(m)={1if  m∈σ⊥0else, \varphi_\sigma(m) = \begin{cases} 1 & \text{if } ~m\in \sigma^\perp \\ 0 & \text{else} \end{cases},

where σ⊥={m∈M ∣ ⟨m,u⟩=0 for all u∈σ}\sigma^\perp = \{m \in M ~\mid~ \langle m,u\rangle = 0 \text{ for all } u \in \sigma\}.

©Isaac Martin. Last modified: January 15, 2024.