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 X\subset Y be a subscheme, and let C_X Y be its normal cone.  We define s(X,Y), the Segre class of X in Y to be s(C_X Y), the Segre class of the normal cone.

It turns out that this is a birational invariant.  Let f:Y'\to Y a morphism of pure-dimensional schemes, X\subset Y and X'=f^{-1}(X), with g:X'\to X, then when f is proper and Y irreducible with each irreducible component of Y' surjectively mapped to Y, we get g_*(s(X',Y'))=\deg(Y'/Y)s(X,Y).  So take f:Y'\to Y birational.  Then \deg(Y'/Y)=1, and so the Segre classes of subschemes push forward to their images.  Additionally, whenever f is flat, we have g^*(s(X,Y))=s(X',Y').

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 X an irreducible subvariety of Y (a variety, not just a scheme here).  Then s(X,Y) is a cycle in A_*(Y).  We define the multiplicity of Y along X (or the algebraic multiplicity of X on Y) to be the coefficient of [X] in s(X,Y), and we denote it by e_X Y.  If we have positive codimension n, then e_X Y[X]=q_*(c_1(\mathscr{O}(1)^n\cap [P(C\oplus 1)])=p_*(c_1(\mathscr{O}(1))^{n-1}\cap [P(C)]) with p,q the projections from P(C), P(C\oplus 1) to X.

Even better, if \tilde{Y} is the blowup of Y at X, and \tilde{X} the exceptional divisor, then e_XY[X]=(-1)^{n-1}p_*(\tilde{X}^n).  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 \mathscr{O}_{X,Y}, which is just n! times the lead coefficient of the polynomial \mathrm{length}_A(A/m^t), where A=\mathscr{O}_{X,Y} and m is the maximal ideal.

Let’s do some quick computations:

Let C be a smooth projective curve of genus g, and C^{(d)} the d-fold symmetric product, which parameterizes the degree d effective divisors.  Let J(C) be the Jacobian, and if we fix p_0\in C, then we get a map u_d:C^{(d)}\to J(C) by D\mapsto D-dp_0.  Now, for d>2g-2, we have that u_d makes this a projective bundle, given by linear systems.  So the first thing we can say is that s(|D|,C^{(d)})=(1+h)^{g-d+r}\cap [|D|], where \dim |D|=r.  For big d, this is simple, for small ones, embed into a big one and the normal bundle restricts nicely.

Now, from g_*(s(X',Y'))=\deg(Y'/Y)s(X,Y), and the above, we can deduce the Riemann-Kempf formula.  For this, we take the image of u_d to be W_d, and we pick a point u_d(D).  We want to know the multiplicity of that point.  Well, we know the class on top is s(|D|,C^{(d)})=(1+h)^{g-d+r}\cap [|D|].  This is just \sum_{i=0}^{g-d+r} \binom{g-d+r}{i}h^i\cap |D|, we then push it forward to p\in W_d, and see that its multiplicity if \binom{g-d+r}{r}, which recovers the Riemann-Kempf formula, and in particular, the classical Riemann Singularity Theorem, when d=g-1.

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