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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 $\tilde{C}$. It \tilde{C}$, which is the a double cover of a conic $D \subset \mathbb{P}^2$; the mathbb{P}^2$. The map $f$ f \colon \tilde{C} \to \mathbb{P}^1$ is a quadruple coverof $\mathbb{P}^1$, and is , obtained by composing $g$ with the projection of $D$ from the point in $\mathbb{P}^2$ corresponding to the unique non-zero section of $L \otimes \mu$.
show/hide this revision's text 2 added 71 characters in body; deleted 2 characters in body; added 6 characters in body

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

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

so

and this shows that the complete linear system $|\phi^*L|$ induces a map $g \colon \tilde{C} \to \mathbb{P}^{n+m-1}$. Then 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 $h^0(L \phi^* H^0(L \otimes \mu)$.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 noting but precisely the canonical map of $\tilde{C}$. It is the double cover of a conic in $\mathbb{P}^2$; D \subset \mathbb{P}^2$; the map $f$ is a quadruple cover of $\mathbb{P}^1$, and is obtained by composing $g$ with the projection of the conic $D$ from the point in $\mathbb{P}^2$ corresponding to the unique section of $h^0(L L \otimes \mu)$.mu$.
show/hide this revision's text 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

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

so the complete linear system $|\phi^*L|$ induces a map $g \colon \tilde{C} \to \mathbb{P}^{n+m-1}$. Then 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 $h^0(L \otimes \mu)$.

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 noting but the canonical map of $\tilde{C}$. It is the double cover of a conic in $\mathbb{P}^2$; the map $f$ is a quadruple cover of $\mathbb{P}^1$, and is obtained by composing $g$ with the projection of the conic from the point in $\mathbb{P}^2$ corresponding to the unique section of $h^0(L \otimes \mu)$.