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:

$D\cdot (\alpha+\alpha')=D\cdot\alpha+D\cdot\alpha'$
$(D+D')\cdot \alpha=D\cdot\alpha+D'\cdot\alpha$
The first two are fairly self explanatory: intersection is distributive on both sides.
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 $g_*(f^*D\cdot \alpha)=D\cdot f_*\alpha$ .
Let $f:X'\to X$ flat, $g$ the appropriate restriction, and we get $f^*D\cdot f^*\alpha=g^*(D\cdot\alpha)$
If $D\sim 0$ then $D\cdot \alpha=0$ .
$D\cdot [D']=D'\cdot[D]$ .

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).

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.

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