Given any triple of positive integers $(g',g,d)$ with $2g'-2\geq d(2g-2)$.


Does there always exist curves $C\_{g'},C\_g$ of genus $g',g$ with a degree $d$ morphism $f\colon C\_{g'}\to C\_g$? 


If we fix $C\_g$ a curve of genus $g$, can we always find a a curve $C\_{g'}$ with genus $g'$ and a degree $d$ morphism $f\colon C\_{g'}\to C\_g$?



Construction of curves and morphisms
Yes. Let me assume for simplicity $g\geq 2$ — the cases $g=0,1$ can be treated in the same way with slight modifications. Put $r=2g'-2-d(2g-2)$, and choose $r+1$ distinct points $p;p\_1,\ldots ,p\_r$ on $C\_g$. The group $\pi:=\pi \_1(C\_g\smallsetminus \{p\_1,\ldots ,p\_r\},p )$ is generated by $2g+r$ elements $a\_1,\ldots ,a\_g;b\_1,\ldots ,b\_g;c\_1,\ldots ,c\_r$, with the relation $c\_1\ldots c\_r\prod (a\_i,b\_i)=1$. Let $\sigma ,\tau $ be two permutations which generate $\mathfrak{S}\_d$. There is a surjective homomorphism $\pi \rightarrow \mathfrak{S}\_d$
which maps each $c\_i$ to the transposition $(1,2)$, $a\_1$ to $\sigma $, $a\_2$ to $\tau $, and all others $a\_i$, $b\_j$ to $1$ (note that $r$ is even). It corresponds to a connected covering $\tilde{C}\rightarrow C\_g $ with a simple ramification point above each $p\_i$, and étale elsewhere. By the Riemann-Hurwitz formula $g(\tilde{C} )=g'$.


