Adjunction formula

The adjunction formula is a very important mathematical tool that helps us relate the geometry of a surface to the geometry of curves on that surface.

Imagine that you are playing with a hula hoop, and someone has drawn some lines on it. Those lines are like curves on a surface. Now, suppose you have a rubber band that you stretch around the hula hoop and it is tight enough to touch all those lines at exactly one point. That point is called the intersection point.

The adjunction formula tells us that the sum of the square of the intersection numbers of each of those curves with itself, minus twice the intersection number of one curve with another, is equal to the degree of the canonical bundle of that surface, which is a fancy way of saying "how twisted up the surface is".

So, if you know how the rubber band stretches around the hula hoop, you can use the adjunction formula to figure out some cool things about the geometry of the curves on the hula hoop. Similarly, if you know how the geometry of the surface is twisted up, you can use the adjunction formula to figure out some things about the curves that live on that surface.

In summary, the adjunction formula helps us understand how the geometry of a surface and its associated curves are related, by using a rubber band analogy.