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).
Pingback: Chern Classes: Part 1 « Rigorous Trivialities