Self-intersection and the normal bundle

Question

Let $X/k$ be a surface nonsingular and proper over an algebraically closed field $k$. Let $C \subset X$ be a nonsingular curve. Then it is clear that the self-intersection $(C \cdot C)_X$ is $\textrm{deg}_C ( \mathcal{N}_{X/C} )$ , basically a matter of definition in intersection theory. More generally, if $X/k$ is a proper variety of dimension $k$, and $Y \subset X$ is a cartier divisor, the the class $[Y\cdot Y] \in A_{k-2}(Y)$ is the class of the line bundle $\mathcal{O}_X(Y) \vert_{Y} = \mathcal{N}_{Y/X}$. Both of these results are fairly easy to prove. I'm asking for something a little different:

$\textbf{Question:}$ I imagine these results are "intuitively clear" at some level to geometers. Let's stick to complex algebraic varieties. In the setting of surfaces, can one explain why the normal bundle controls the number of points that divisors linearly equivalent to $C$ meet $C$? I want to say that this "follows" because we can consider the normal bundle as a "tubular neighborhood", but I don't know how to do this precisely, or how to finish the argument. How about in the higher dimensional case?

Answer 1

I'd like to expand a bit on the excellent comments of Charles Staats and Donu Arapura. They both suggest understanding the self-intersection number of a curve as the number of fixed points of an infinitesimal deformation of the curve, which is manifestly the degree of the normal bundle when such a deformation exists. Here's a slightly more pedestrian route, which I think has the benefit of being rigorous and almost as intuitive.

Suppose we have two curves in a surface: $$\iota_C: C\hookrightarrow X, \iota_D: D\hookrightarrow X.$$ If $C\cap D$ has dimension zero, the intersection number should manifestly be $$C\cdot D:=\dim\Gamma(C\cap D, \mathcal{O}_{C\cap D})=\dim\Gamma(C, \iota_C^*\mathcal{O}_D)=\dim\Gamma(C, \iota_C^*\mathcal{O}_D(D)).$$ (The twist in the last equality does nothing by our assumption on the dimension of $C\cap D$ --- I've just inserted it to simplify things slightly later on.) We'd like to write this as an Euler characteristic, to make it constant if we vary $C$ or $D$ in a flat family. But this is easy; since $\mathcal{O}_{C\cap D}$ has zero-dimensional support, it has no higher cohomology, so its Euler characteristic equals $C\cdot D$ as defined above. Line bundles are nice (and more importantly, are acyclic with respect to restriction), so we use the short exact sequence $$0\to \mathcal{O}_X\to \mathcal{O}_X(D)\to \mathcal{O}_D(D)\to 0$$ to rewrite this Euler characteristic as $$\chi(\mathcal{O}_X(D)|_C)-\chi(\mathcal{O}_C)=\operatorname{deg}(\mathcal{O}_X(D)|_C).$$

I think this is a reasonably intuitive motivation for the definition of the intersection number. So to fully answer your question, one should give an intuitive reason for why $\mathcal{O}_X(D)|_D$ is $\mathcal{N}_{D/X}.$ Of course, this is just the definition of the normal bundle, but let's motivate the definition. First, why is the conormal bundle $I/I^2$, for $I$ the ideal sheaf of a closed subvariety $V\subset X$? Well, an element of $I/I^2$ is precisely a function on $X$ vanishing at $V$, but ignoring higher-order terms. A section to the normal bundle precisely takes functions $f$ defined in a neighborhood of $Y$ and differentiates them--but the partial derivative should depend only on the first-order part of $f$. So the normal bundle should be precisely $(I/I^2)^\vee$. This is another name for $\mathcal{O}_X(D)|_D.$

I hope that was some reasonable intuition/motivation.