Actually calculating an Intersection of Variety and Divisor

Question

This has been bugging me for a while now.

Say I have a projective variety given by some polynomial $P$ and the canonical divisor of the projective space. How can I concretly calculate the Intersection of the two?

And by concretly I mean, actually get a result that is not purely abstract? (Like actual Intersection points, degree, etc...)

Answer 1

Unfortunately, you will run into a basic difficulty in your desire to make this example concrete, which is that the canonical bundle on projective space is anti-ample, and so the canonical bundle is not effective. (Rather, it is its dual that is an effective divisor.)

More precisely, if $H$ is the linear equivalence class of a hyperplane in $\mathbb P^n$, then the canonical divisor $K$ is equal to $-(n+1)H$.

So if you want to intersect $K$ with $V(P)$ (the variety cut out by the polynomial $P$) you will have to use at least a little bit of theory, even if only to interpret what you are doing.

You might be better of starting with $H$ itself. Then $H \cap V(P)$ is the linear equivalence class of a hyperplane section of $V(P)$. Assuming that $V(P)$ is smooth, then Bertini's theorem says that a generic member of the linear equivalence class $H\cap V(P)$ will be smooth, and even irreducible if the dimension of $V(P)$ is at least two (i.e. if $n \geq 3$).

Then one way to write $K \cap V(P)$ is simply as $-(n+1) \bigl( H \cap V(P) \bigr)$.

Alternatively, one could consider the linear equivalence class $(n+1)\bigl( H \cap V(P)\bigr).$ This is the class of intersections of $V(P)$ with a degree $n+1$ hypersurface, and again a generic member is smooth (and irreducible if $V(P)$ is of dimension at least $2$). Then you think of $K \cap V(P)$ as being the negative of this class.

Answer 2

This is more of a comment than an answer (but I cannot comment yet). Isn't it the case that "the" canonical divisor is any divisor representing the linear equivalence class of the canonical bundle? We would get intersection points only after choosing a representative (and coordinates), and the points may depend on our choice. Only the intersection class (in a Chow ring) should be intrinsic, I think. I would appreciate hearing more on this too!