Today we start actually performing intersections.  Fix X a scheme, j:V\to X an inclusion of a subvariety, \dim X=n, \dim V=k, and let D be a divisor on X.  The big definition for today: D\cdot [V]=[j^*(D)] in A_{k-1}(|D|\cap V) where |D| is the support.

More generally, let \alpha=\sum n_V[V] be a cycle, then D\cdot \alpha=\sum n_V D\cdot [V] is the intersection class.  Let’s go through a list of all sorts of nice properties that this intersection product satisfies:

  1. D\cdot (\alpha+\alpha')=D\cdot\alpha+D\cdot\alpha'
  2. (D+D')\cdot \alpha=D\cdot\alpha+D'\cdot\alpha
    The first two are fairly self explanatory: intersection is distributive on both sides.
  3. Let f:X'\to X proper, D a divisor on X and \alpha a cycle on X', and set g to be the restriction of f to f^{-1}(|D|\cap|\alpha|).  Then we get the Projection Formula g_*(f^*D\cdot \alpha)=D\cdot f_*\alpha.
  4. Let f:X'\to X flat, g the appropriate restriction, and we get f^*D\cdot f^*\alpha=g^*(D\cdot\alpha)
  5. If D\sim 0 then D\cdot \alpha=0.
  6. D\cdot [D']=D'\cdot[D].

Now, what does this get us? Look in \mathbb{P}^2, and let C and C' be curves with no common components, of degrees d and e.  Because \mathbb{P}^2 is a surface, we can view C and D both as being divisors and being 1-cycles.  So, we look at C\cdot [C'].  This is going to be a zero cycle on C'\cap C, and so we look at \int_{C\cap C'}C\cdot[C'].   We want to compute this number.  Now, we make use of equivalence.  A divisor is linearly equivalent to d\ell, where \ell is a line, so we have d\int_{\ell\cap C'}\ell\cdot[C'].  However, by symmetry, we have d\int_{\ell\cap C'}C'\cdot[\ell], and using linear equivalence again, we get de\int_{\ell\cap \ell}\ell\cdot[\ell], which is easily seen as de.

So our new formalism contains Bezout’s Theorem, and generalizes it: let D_1,\ldots,D_n be divisors in \mathbb{P}^n, then \int D_1\cdot\ldots D_n=d_1\ldots d_n where d_i=\deg D_i.  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).

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