Regular map (algebraic geometry)

Okay, so let's talk about maps. A map is like a big picture that shows you where things are. For example, a map of your house might show you where your bedroom, the kitchen, and the bathroom are.

Now let's talk about a regular map in algebraic geometry. Algebraic geometry is kind of like making maps for things that are made of numbers. Instead of showing where rooms are in a house, we might be looking at something called a "variety." A variety is a special kind of mathematical object that has a set of solutions to a particular set of equations.

A regular map in algebraic geometry is like a map between two varieties. It shows you how to match up points on one variety with points on the other variety. But there's a catch - the map has to be "regular."

What does regular mean? Think of it like this: imagine you have a toy bucket with all different shaped blocks inside. Some blocks might be round, some might be square, and some might have weird angles. If you tried to put all those blocks through a small round hole, some of them might get stuck or not fit in properly.

A regular map is like a hole that all the blocks can pass through without getting stuck. It's a nice, smooth, even hole that matches up with all the different shapes of blocks.

So a regular map in algebraic geometry is like a nice, smooth, even picture that shows you how to match up points on one variety with points on another variety. And just like we want our maps to show us where things are without getting lost or confused, we want our regular maps in algebraic geometry to be nice and clear and easy to follow.