Let $X$ be a smooth projective algebraic curve over $\mathbb{C}$ (or equivalently a compact Riemann surface). Let $d$ be any positive integer.
Does there exist a smooth projective curve $Y$ and a finite morphism $f : Y\rightarrow X$, such that $f$ has degree $d$, where by $degree$, I mean the degree of the extension of function fields $[K(X):K(Y)]$, or the degree of $f$ as a ramified covering map, if one prefers to think them as Riemann surfaces.
Thanks in advance.
Yes. Let $X \subset \Bbb P^n$ be your curve, where the homogenous coordinates are $z_0, \dots, z_n$. Let $H$ be an hyperplane given by $\ell = 0$, which do not contain $X$.
Now consider the weighted projective space $\Bbb P(d,d \dots, d, 1)$ with coordinates $z_0, \dots, z_n, w$ and the variety $ Z$ given by $w^d = \ell$. There is a natural map $f : Z \to \Bbb P^n$ which forgets the $w$ coordinate, and $f^{-1}(X)$ is by construction a $d$-fold cover of $Y$, precisely branched at $X \cap H$.