So, last time we talked about Segre classes and cones. Now, we’re going to move ahead, and talk about a specific cone in detail, the Normal cone we defined on Monday. Let be a subscheme, and let
be its normal cone. We define
, the Segre class of
in
to be
, the Segre class of the normal cone.
It turns out that this is a birational invariant. Let a morphism of pure-dimensional schemes,
and
, with
, then when
is proper and
irreducible with each irreducible component of
surjectively mapped to
, we get
. So take
birational. Then
, and so the Segre classes of subschemes push forward to their images. Additionally, whenever
is flat, we have
.
So the Segre classes behave really nicely with respect to the functorial maps we have. So we can start using it to define other things, and we might even be able to compute them by pushing around into simple cases, and then pulling back to our case.
Now, take an irreducible subvariety of
(a variety, not just a scheme here). Then
is a cycle in
. We define the multiplicity of
along
(or the algebraic multiplicity of
on
) to be the coefficient of
in
, and we denote it by
. If we have positive codimension
, then
with
the projections from
to
.
Even better, if is the blowup of
at
, and
the exceptional divisor, then
. So we can move the problem to being intersecting a divisor with itself a bunch of times, pushing forward, and then checking a sign. This turns out to be the same as the definition of the multiplicity of the local ring
, which is just
times the lead coefficient of the polynomial
, where
and
is the maximal ideal.
Let’s do some quick computations:
Let be a smooth projective curve of genus
, and
the
-fold symmetric product, which parameterizes the degree
effective divisors. Let
be the Jacobian, and if we fix
, then we get a map
by
. Now, for
, we have that
makes this a projective bundle, given by linear systems. So the first thing we can say is that
, where
. For big
, this is simple, for small ones, embed into a big one and the normal bundle restricts nicely.
Now, from , and the above, we can deduce the Riemann-Kempf formula. For this, we take the image of
to be
, and we pick a point
. We want to know the multiplicity of that point. Well, we know the class on top is
. This is just
, we then push it forward to
, and see that its multiplicity if
, which recovers the Riemann-Kempf formula, and in particular, the classical Riemann Singularity Theorem, when
.
Just a question – why do you restrict to varieties starting in paragraph 4? If X is an irreducible closed subscheme can’t you still make the same definition and have (some of) the same properties? If it’s not irreducible one can make the definitions of multiplicity along an irreducible component.
Yeah, it’ll probably mostly work out. Mostly I’m figuring that because we’re going to use normal cones and Segre classes to define the intersection product, and the varieties are a basis for the Chow ring, that’s all that we’re going to need, so we can work in the simpler case if we like (and generally, I prefer varieties over schemes).