Question

Let $C$ be a curve. Then I know of two ways to create morphisms. To get morphisms from $C$, take a line bundle of any degree $L$ and use the linear system it determines to get a map into projective space, which may or may not be injective, so you get a map to another curve. To get a morphism to $C$, a particular case is to take a line bundle $\mu$ of order 2, which gives us a curve $\tilde{C}$ as a double cover of $C$.

How do these two methods interact? For instance, given a curve $C$, and the tower $\tilde{C}\to C\to \bar{C}$ with the first map given as a double cover by $\mu$, and the second map given by a line bundle $L$, presumably the geometry of this tower is connected to the line bundle $L\otimes\mu$, but it's not obvious to me how to connect the two notions to get any actual information.

Answer 1

A possible answer is the following. Assume

$h^0(L)=n, \quad h^0(L \otimes \mu)=m$,

set

$\phi \colon \tilde{C} \to C, \quad \psi \colon C \to \bar{C} \subset \mathbb{P}^{n-1}$

and let $f=\psi \circ \phi \colon \tilde{C} \to \mathbb{P}^{n-1}$ be the composition.

Then $f^*\mathcal{O}_{P^n}(1)=\phi^*L$. On the other hand, by projection formula we have

$h^0(\phi^*L)=h^0(L) + h^0(L \otimes \mu)=n+m$,

and this shows that the complete linear system $|\phi^*L|$ induces a map $g \colon \tilde{C} \to \mathbb{P}^{n+m-1}$. Moreover, the map $f$ is obtained by composing $g$ with the projection from the linear subspace $\mathbb{P}^{m-1} \subset \mathbb{P}^{n+m-1}$ corresponding to the natural inclusion $\phi^* H^0(L \otimes \mu) \subset H^0(\phi^*L)$.

The easiest example is perhaps the following. Assume that $C$ is a genus $2$ curve and take $L=K_C$. Then $\bar{C}=\mathbb{P}^1$, $\tilde{C}$ is a hyperelliptic (this can be proven) genus $3$ curve and $\phi^*L=K_{\tilde{C}}$. We have

$h^0(L)=2, \quad h^0(L \otimes \mu)=1$

and the map $g \colon \tilde{C} \to \mathbb{P}^2$ is precisely the canonical map of $\tilde{C}$, which is a double cover of a conic $D \subset \mathbb{P}^2$. The map $f \colon \tilde{C} \to \mathbb{P}^1$ is a quadruple cover, obtained by composing $g$ with the projection of $D$ from the point in $\mathbb{P}^2$ corresponding to the non-zero section of $L \otimes \mu$.