---
date: 2023-03-01 21:27
title: Toric Varieties
aliases:
  - Toric Varieties
flashcard: Reading::Unsorted
created: 2023-03-30T13:11
updated: 2024-01-05T21:48
---

# Toric Varieties

## Essentials


- [x] When is a toric variety $X_\Sigma$ **normal**? ✅ 2023-03-30
	- [ ] $\spec \CC[S]$ is normal for $S\leq \Lambda$ a submonoid iff $S$ is saturated in $\Lambda$.
- [x] When is $X_P$ normal? ✅ 2023-03-30
	- [ ] Iff $S_m\da \gens{m' - m \st m' \in P \intersect M}_{\NN}$ is saturated in $M$.
- [x] When is a toric variety $X_\Sigma$ **smooth**? ✅ 2023-03-30
	- [ ] $\rho_i$ extend to a $\ZZ\dash$basis of $\Lambda$
- [x] When is $X_P$ smooth? ✅ 2023-03-30
	- [ ] $X_{\Sigma}$ is smooth where $\Sigma$ is the normal fan of $P$.
	- [ ] Equivalently the primitive edge vectors at each vertex form a full $\ZZ\dash$basis of $M$.
- [x] When is $P$ ample? ✅ 2023-03-30
	- [ ] $S_m \da \gens{P\intersect M  - m}_{\NN}$ is saturated in $M$.

## Notation

- [x] What are $H_{u, b}$ and $H_{u,b}^+$ for $u\in N_\RR$? ✅ 2023-03-30
	- [ ] $$H_{u, b}=\left\{m \in M_{\mathbb{R}}:\langle m, u\rangle=b\right\}, \quad H_{u, b}^{+}=\left\{m \in M_{\mathbb{R}}:\langle m, u\rangle \geq b\right\} \subset M_\RR$$
- [x] What is $X_P$? ✅ 2023-03-30
	- [ ] For $A \da P \intersect M = \ts{m_0,\cdots, m_s}$, get $T\actson \PP^n$ for $T$ the character lattice of $M$ by $t.\vector x = \tv{\chi^{m_0}(t) \cdot x_0: \cdots : \chi^{m_s}(t) \cdot x_s}$ and define $X_A \da \cl_{\PP^s}(T. \tv{1:\cdots:1})$.
- [x] What is $u_F$? ✅ 2023-03-30
	- [ ] For $P \subset M$, $u_F\in N_\RR$ is orthogonal to the face $F$.
- [x] What is the polar dual $P^*$? ✅ 2023-03-30
	- [ ] $P^*=\left\{u \in N_{\mathbb{R}}:\langle m, u\rangle \geq-1 \text { for any } m \in P\right\} \text {. }$