## 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).