Zariski's lemma
Okay, so let's say you have a big field where you can play with numbers like adding, subtracting, multiplying and dividing. And let's say you have a magic wand that can turn any of these numbers into a shape like a point, a line or a curve.
Now, when you put some of these shapes together, they form a bigger shape, like a square or a circle. But sometimes, even if you put a lot of shapes together, they don't form any bigger shape at all. These are called "disconnected shapes".
But the cool thing is that you can use some magic to figure out which shapes can be connected and which cannot. This is what Zariski's Lemma does. It tells you that, if you have a certain kind of shape made out of points, lines and curves, and it's disconnected, then you can find some other shapes that, when you put them together with the first one, they form a bigger shape that is connected. And you can keep doing this until you cannot make any more connections!
So basically, Zariski's Lemma is a magic tool that helps you see how shapes made out of numbers are connected. It's like putting a puzzle together, but with shapes instead of pieces, and you can always make the puzzle bigger and connected if you want to.
Now, when you put some of these shapes together, they form a bigger shape, like a square or a circle. But sometimes, even if you put a lot of shapes together, they don't form any bigger shape at all. These are called "disconnected shapes".
But the cool thing is that you can use some magic to figure out which shapes can be connected and which cannot. This is what Zariski's Lemma does. It tells you that, if you have a certain kind of shape made out of points, lines and curves, and it's disconnected, then you can find some other shapes that, when you put them together with the first one, they form a bigger shape that is connected. And you can keep doing this until you cannot make any more connections!
So basically, Zariski's Lemma is a magic tool that helps you see how shapes made out of numbers are connected. It's like putting a puzzle together, but with shapes instead of pieces, and you can always make the puzzle bigger and connected if you want to.