---
created: 2024-05-03T00:13
updated: 2024-06-06T12:04
---

A **normalization** of an irreducible variety $X$ is an irreducible normal variety $\widetilde{X}$, together with a regular map $\nu: \widetilde{X} \rightarrow X$ which is a finite birational equivalence. A $\operatorname{map} \varphi: X \rightarrow X^{\prime}$ between varieties is **finite**, if any point $x \in X^{\prime}$ has an affine neighborhood $\mathcal{V}$ such that the preimage $\mathcal{U}:=\varphi^{-1}(\mathcal{V})$ is affine and the restriction $\varphi: \mathcal{U} \rightarrow \mathcal{V}$ is a **finite map**, that is every point has a finite number of preimages.

Definition 5.4.1. A **(convex polyhedral) cone** in $\mathbb{R}^n$ is a set of the form
$$
C=\left\{a_1 v_1+\ldots+a_r v_r \in \mathbb{R}^n \mid a_1, \ldots, a_r \geq 0\right\}
$$
for some finite set of vectors $v_1, \ldots, v_r \in \mathbb{R}^n$, then called the **generators** of the cone $C$. The cone $C$ is **rational** if it admits a set of generators in $\mathbb{Z}^n$. The cone $C$ is **smooth** if it admits a set of generators which is part of some $\mathbb{Z}$-basis of $\mathbb{Z}^n$.

The **toric variety $X_{\mathcal{F}}$ associated to a fan $\mathcal{F}$ in $\mathbb{R}^n$** is the result of gluing the affine toric varieties $X_C:=\operatorname{Spec}_{\mathrm{m}} \mathbb{C}\left[S_C\right]$ (for all $C \in \mathcal{F}$ ) by identifying $X_C$ with the correponding Zariski open subset in $X_{C^{\prime}}$ whenever $C$ is a face of $C^{\prime}$.

Some cornerstones of my work:

- Real reflection group and Coxeter diagrams (rep theory)
- Lattice theory (NT)
- Tilings of hyperbolic spaces (geometry/general audience)
- Hermitian symmetric spaces (geometry/topology)
- Toric geometry (AG)
- Moduli spaces (AG)

I should dig up and compile all of my actual examples of computations with toric varieties.

## Basic review

- [ ] Completeness
- [ ] Properness
- [ ] The three valuative criteria
- [ ] Closed immersions
- [x] Separated ✅ 2024-06-06
	- [ ] The diagonal is a closed immersion.

## Exercises

![[Pasted image 20230819164238.png]]