Relative effective Cartier divisor
Okay kiddo, let's talk about "relative effective Cartier divisors". This is a really tough topic, so I'll try to explain it in a way that makes sense for you!
Imagine you have a big piece of paper with a shape drawn on it, like a circle or a square. This is called a "variety". Now, pretend that you want to draw a line on that paper that goes right through the middle of the shape. That line is called a "divisor".
Sometimes, that line might pass through some points on the shape and not others. The points where it passes through are called "intersection points". If you drew that line carefully, you might notice that it only intersects with a few points on the shape, not all of them.
Now, let's say that we only care about the points where that line intersects the shape. We don't care about the rest of the shape that the line does not touch. We can consider the part of the shape that the line touches to be a "divisor on the variety".
Sometimes, we might want to look at the "effective" part of that divisor. This means the part that is not made up of any negative numbers. In other words, we only look at the points where the line intersects the shape in a positive way, not a negative way.
Now, if we want to look at the "relative effective Cartier divisor", that means we're looking at the part of the shape that the line intersects, but only in relation to some other part of the shape.
Whew, that was a lot to explain! But essentially, a relative effective Cartier divisor is a special way to look at a line drawn on a shape, and only consider the parts where it intersects in a positive way, and only in relation to some other part of the shape.
Imagine you have a big piece of paper with a shape drawn on it, like a circle or a square. This is called a "variety". Now, pretend that you want to draw a line on that paper that goes right through the middle of the shape. That line is called a "divisor".
Sometimes, that line might pass through some points on the shape and not others. The points where it passes through are called "intersection points". If you drew that line carefully, you might notice that it only intersects with a few points on the shape, not all of them.
Now, let's say that we only care about the points where that line intersects the shape. We don't care about the rest of the shape that the line does not touch. We can consider the part of the shape that the line touches to be a "divisor on the variety".
Sometimes, we might want to look at the "effective" part of that divisor. This means the part that is not made up of any negative numbers. In other words, we only look at the points where the line intersects the shape in a positive way, not a negative way.
Now, if we want to look at the "relative effective Cartier divisor", that means we're looking at the part of the shape that the line intersects, but only in relation to some other part of the shape.
Whew, that was a lot to explain! But essentially, a relative effective Cartier divisor is a special way to look at a line drawn on a shape, and only consider the parts where it intersects in a positive way, and only in relation to some other part of the shape.