See my two previous questions here: Intuition for thinking about R-module of Kähler differentials, universal receptacles, derivations? and Kähler differentials, define valuation? for background.
If $\omega \in \Omega_{K_C/\mathbb{C}}$, then we define $$\text{div}(\omega) := \sum_{p \in C} v_p(\omega)p,$$so we can associate divisors to differentials.
Suppose $C \subset \mathbb{CP}^2$ is defined by $F(x, y, z) = 0$, and $C \cap \{z = 0\}$ is finite. $C - (C \cap \{z = 0\})$ is the affine curve defined by $f(x, y) = F(x, y, 1)$. What is $\text{div}(dx)$? If $p \in [x_0, y_0, 1]$, then $v_p(dx) = 0$ if $x - x_0$ is a uniformizing parameter at $p$. We have that $x - x_0$ is a uniformizing parameter as long as $f_y(p) \neq 0$.
Any divisor $D$ which is linearly equivalent to $\text{div}(\omega)$, i.e. $D \sim \text{div}(\omega)$, is also the divisor of a differential, trivially. Any divisor of this form is called a canonical divisor. In fact, we see that all divisors of this form are linearly equivalent, so the canonical divisors are an equivalence class in the class group.
(The class group $\text{Cl}(C)$ is defined by the exact sequence $0 \to \mathbb{C}^\times \to K_C^\times \to \text{Div}(C) \to \text{Cl}(C) \to 0.$)
My two questions are as follows.
- What is the intuitive way/motivation behind thinking about $\text{div}(\omega)$ (defined as above) in context of algebraic curves, preferably geometrically?
- What is the intuitive way/motivation behind thinking about canonical divisors (defined as above) in context of algebraic curves, preferably geometrically?
Thanks in advance.
