Zariski's main theorem

Zariski's main theorem is a big idea in math that helps us understand what kinds of objects we can use to describe spaces that we study in algebraic geometry.

Imagine you have a big field that is full of flowers, and you want to understand what shape the field is. You can't walk through all of the flowers one by one to figure out the shape, but if you draw a map of the field, you can get a pretty good idea.

In algebraic geometry, we are interested in understanding shapes that can be described by equations. Imagine you have a shape in space that is determined by a bunch of equations. For example, if you have a 3-dimensional space, then your shape might be determined by a whole bunch of equations that have three variables each.

Zariski's main theorem tells us that we can study this shape by looking at a smaller shape that is described by simpler equations. This smaller shape is called the "regular locus," and it's basically the part of the original shape where the equations behave nicely and don't have any weird holes or singularities.

So, just like the map of the field helps us understand the shape of the field without having to walk through all the flowers, the regular locus helps us understand the shape described by the equations without having to study the details of each individual equation.

Zariski's main theorem is like a guide to help us find the regular locus, and it tells us that the regular locus is a very important part of the shape that we are studying. By understanding the regular locus, we can learn a lot about the original shape, even if it's very complicated and hard to understand.