Let $X \to \operatorname{Spec}(k)$ be a proper curve over the field $k$. I want to show that there is some effective Cartier divisor $D \geq 0$ on $X$ such that $\operatorname{supp}(D) \cap X_i \neq \emptyset$ where $X_i$ are the irreducible components of $X$.
My idea is to come up with effective divisors on the components which agree on the overlaps to glue these together. Thus for me this sounds like
- Show the existence of effective divisors on the components
- and then show (with some kind of approximation argument) that these can be chosen to fit together.
But maybe all this follows from a more elegant concept. I appreciate any kind of help or hints!