Today we start actually performing intersections. Fix a scheme, an inclusion of a subvariety, , and let be a divisor on . The big definition for today: in where is the support.
More generally, let be a cycle, then is the intersection class. Let’s go through a list of all sorts of nice properties that this intersection product satisfies:
The first two are fairly self explanatory: intersection is distributive on both sides.
- Let proper, a divisor on and a cycle on , and set to be the restriction of to . Then we get the Projection Formula .
- Let flat, the appropriate restriction, and we get
- If then .
Now, what does this get us? Look in , and let and be curves with no common components, of degrees and . Because is a surface, we can view and both as being divisors and being -cycles. So, we look at . This is going to be a zero cycle on , and so we look at . We want to compute this number. Now, we make use of equivalence. A divisor is linearly equivalent to , where is a line, so we have . However, by symmetry, we have , and using linear equivalence again, we get , which is easily seen as .
So our new formalism contains Bezout’s Theorem, and generalizes it: let be divisors in , then where . So this formalism should be useful as a beginning to intersection theory proper. That seems like a good place to stop, so either there will be a second post later today, or a longer one tomorrow (or two tomorrow).