Nonsingular curve $C$ of degree 4, exists rational function $f: C \to \mathbb{CP}^1$ of degree 2? Suppose $C \subset \mathbb{CP}^2$ is a nonsingular curve of degree $4$. Does there exist a rational function $f: C \to \mathbb{CP}^1$ of degree $2$?
 A: We show that there does not exist such a rational function.
By Riemann-Roch, we have that$$\ell(K - p) = 1 + \ell(p)$$and$$\ell(K - p - q) = \ell(p + q).$$We know that$$\ell(p) = 1,$$since $p$ is an effective divisor, and no meromorphic function on a compact Riemann surface can have just $1$ simple pole, so we have that$$\ell(K - p) = 2.$$Hence, $\ell(K - p - q)$ is either $1$ or $2$.
If$$\ell(K - p - q) = 2,$$choose a rational function from it, $f$, which is nonconstant. This induces a degree $2$ map into $\mathbb{CP}^1$, so this is equivalent to saying that $C$ is hyperelliptic. But the canonical divisor of a hyperelliptic curve is not very ample, whereas $K_C$ is; this follows immediately from the adjunction formula and this link. $($Sans machinery, we can also write out explicitly a basis for the canonical linear system and prove that the canonical map is an embedding, but this is messy and basically equivalent.$)$
$\mathbb{C} \subset L(p + q)$ by effectiveness, so it is precisely $\mathbb{C}$. Any rational function $f: C \to \mathbb{CP}^1$ of degree $2$ would have $2$ poles, and hence, fall in $L(p + q)$ but be nonconstant, which is a contradiction.
A: Not necessarily. Most genus $3$ curves are not hyperelliptic.
Quoting wikipedia :

This is seen heuristically by a moduli space dimension check. Counting constants, with n = 2g + 2, the collection of n points subject to the action of the automorphisms of the projective line has (2g + 2) − 3 degrees of freedom, which is less than 3g − 3, the number of moduli of a curve of genus g, unless g is 2.

