Definition of the degree of a sheaf

Question

Let $C$ and $D$ be smooth closed curves on a projective smooth surface $X$ of finite type over an algebraically closed field.

I'm looking for a definition of the term $\deg_C(\mathcal{O}(D)_{|C})$. This counts the number of points in the intersection of $C$ and $D$ (provided intersect transversally).

Is this the dimension of the global sections of the the sheaf $\mathcal{O}_{C\cap D}$? Is there a more general definition of $\deg \mathcal{F}$ for a coherent sheaf $\mathcal{F}$ over some scheme $X$?

thanks in advance!

Answer 1

If $C,D$ are completely arbitrary divisors on a smooth projective surface defined over an algebraically closed field $k$, one can define their intersection number, a rational integer $$\langle C,D\rangle\in \mathbb Z$$ and obtain a bilinear map $$Cl(S)\times Cl(S)\to \mathbb Z $$ If $C,D$ are effective and have no common component we have $$\langle C,D\rangle=h^0(S,\mathcal O_{C\cap D})=\sum_{s\in C\cap D} \operatorname {dim} \mathcal O_{S,s}/(f_s,g_s)$$ where $C\cap D$ is the scheme theoretic intersection and $f_s,g_s$ are local equations for $C,D$ at $s$.
And finally if $C$ is a prime divisor and $D$ is arbitrary, we have $$\langle C,D\rangle=h^0(C,\mathcal O_S(D)\vert C)$$