Toric variety

Okay, so imagine you have a map of the world, right? And on this map, you can see all the countries and their shapes. Now, imagine you want to figure out which points on this map are really important and special. Maybe they are the capitals of the countries, or maybe they are places where lots of people go to visit.

Now, let's imagine that each of these special points on the map is represented by a dot. But these dots aren't just random - they are arranged in a special way that tells us something about the shape of the countries around them. Some of the dots might be closer together, indicating that the countries around them have similar shapes, while others might be farther apart, indicating that the countries around them are very different.

A toric variety is kind of like this map, but instead of representing countries and capitals, it represents something called an algebraic variety. This is a special kind of shape that we can study using algebra, which is like math but with letters and symbols instead of numbers.

In a toric variety, the dots that represent important points are called "torus orbits". These orbits are arranged in a special way that reflects the shape of the algebraic variety. For example, if the algebraic variety has lots of curves and lines in it, the torus orbits might be arranged in a grid-like pattern, like the streets on a city map.

So, to sum it up: a toric variety is like a map that represents a special kind of shape using dots arranged in a special way. The dots tell us important things about the shape of the variety, like where to find important points or how different parts of the variety are related to each other.