How to define the intersection product of a curve with itself: $(C\cdot C)$?

Question

I'm reading through Positivity in Algebraic Geometry by Lazarsfeld, and it defines the intersection product for bundles $L$ and (Cartier) divisors $D$ on a complete, irreducible complex variety $X$, via chern classes (the specifics of the definition are not necessarily relevant to this question).

However, in (for example) in exercise 1.2.31, it mentions the intersection product of an irreducible curve on a smooth projective surface with itself: $(C\cdot C)$. How should one interpret this? It also makes mention of the sheaf $O_X(mC)$, but I am not sure what this entails either -- it seems neither to be a twisted sheaf nor a sheaf $O_X(mD)$ for a divisor $D$, again unless we can consider $C$ as a Cartier divisor.

Questions: Am I supposed to view $C$ as a Cartier divisor here? If so, how? Specifically, how should I define $(C\cdot C)$ and $O_X(mC)$?

Answer 1

First a disclaimer: I'm a bit rusty with algebraic geometry, so apologies for inaccuracies.

An irreducible curve on a smooth projective surface is by definition a prime divisor which corresponds to an effective Cartier divisor since the surface is nice enough (it is smooth). In particular, you view $\mathcal{O}_X(mC)$ as the line bundle of this divisor $mC$ (see e.g. Vakil or Hartshorne for a definition).

The intersection product can e.g. be defined as $(C \cdot C) = \deg{\mathcal{O}_S(C)|_C}$ where $C \subseteq S$ is the curve on the surface. There is also a more general notion of the intersection product of invertible sheaves:

Definition. Let $\mathcal{F} \in \mathbf{Coh}_X$ be a coherent sheaf with proper support of dimension $\leq n$ over a projective $k$-variety and $\mathcal{L}_1, \dots, \mathcal{L}_n$ be invertible sheaves on $X$. Then, $$ (\mathcal{L}_1 \cdot \mathcal{L_2} \cdot \cdots \cdot \mathcal{L_n} \cdot \mathcal{F}) = \sum_{\{i_1, \dots, i_m \} \subseteq \{1,\dots,n \}} (-1)^m \chi(X, \mathcal{L}_{i_1}^{\vee} \otimes \dots \otimes \mathcal{L}_{i_m}^{\vee} \otimes \mathcal{F})$$ is called intersection of $\mathcal{L}_1, \dots, \mathcal{L}_n$ with $\mathcal{F}$.

For effective Cartier divisors $C,D$ on $X$, we interpret/define $$C \cdot D = (C \cdot D) = (\mathcal{O}_X(C) \cdot \mathcal{O}_X(D) \cdot \mathcal{O}_X).$$ You can e.g. find this viewpoint wonderfully written in Vakil's Rising Sea, chapter 20. In that regard, the fact $(C \cdot C) = \deg{\mathcal{O}_S(C)|_C}$ appears as Exercise 20.2.A.