Intersections with Divisors

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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About Charles Siegel

Charles Siegel is currently a postdoc at Kavli IPMU in Japan. He works on the geometry of the moduli space of curves.
This entry was posted in Intersection Theory, MaBloWriMo. Bookmark the permalink.

One Response to Intersections with Divisors

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

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