Possible degrees of nonconstant map $f:C\rightarrow \mathbb{P}^1$ for a plane curve $C$ I am looking for the possible degrees of nonconstant map $f:C\rightarrow \mathbb{P}^1$ for a plane curve $C$. By combining the Brill-Noether theorem with the equality $g={d-1\choose 2}$ for a plane curve of genus $g$ and degree $d$, one can get some inequalities, for example that such a map does not exist of degree less than or equal to $d-2$.
However, I feel there should be a more elmentary way to approach this problem that gives better bounds, for example by directly considering $g_d^r$ and divisors.
Question: What are the admissable degrees of a nonconstant map $f: C\rightarrow \mathbb{P}^1$? How does one elementarily approach this problem?
 A: Here's a more elementary proof that no smooth plane curve of degree $d>2$ has a $g_e^1$ for any $e<d-1$ (where we use the 'strict' definition of a $g_e^1$ as a divisor $D$ with $\deg D = e$ and $|D|=1$, and one should add some minor hypotheses on $e$ if one is working in positive characteristic). $\def\PP{\mathbb{P}}\def\cO{\mathcal{O}}\def\cI{\mathcal{I}}\def\G{\Gamma}$
We'll need a more geometric interpretation of Riemann-Roch for this: starting from $l(D)-l(K-D)=\deg D+1-g$, we can rewrite this as $l(D)-1 = \deg D-1-g+l(K-D)+1$.
It's clear we can interpret $l(D)-1$ as $\dim |D|$, and we can also interpret $-g+1+l(K-D)$ as $-(g-(l(K-D)-1))$, the negative of the dimension of the space of hyperplanes in $\PP^{g-1}$ which vanish on the image of $D$ under the canonical embedding.
So Riemann-Roch also says that $\dim |D|=\deg D-1-\dim \overline{\varphi_K(D)}$, where the last term is the dimension of the image of $D$ under the canonical embedding (suitably construed when $D$ contains points of the form $nP$).
This says that $|D|$ is equal to the difference between the 'expected' dimension of the span of $\deg D$ points and the dimension of the actual span of $\deg D$ points under $\varphi_K$, the canonical embedding.
In particular, if we have a $g_e^1$, this says that there are $e$ (distinct) points on the canonical curve which lie in an $(e-2)$-plane: we assume that $e$ is such that the map induced by the $g_e^1$ is separable, and therefore generically unramified.
We'll show that if $e<d-1$ this cannot happen by finding a section of the canonical bundle which vanishes on any $e-1$ of these points but not the $e^{th}$.
For a smooth plane curve $X$ of degree $d$, the canonical bundle is $\cO_X(d-3)$.
It's enough to find a section of $\cO_{\PP^2}(d-3)$ which satisfies our vanishing/nonvanishing conditions: the restriction map on global sections is injective because the kernel is $\G(\cI_X(d-3))\cong \G(\cO_{\PP^2}(-d+d-3)) \cong \G(\cO_{\PP^2}(-3))=0$.
If $e<d-1$, then $e-1\leq d-3$, and it suffices to solve the problem for $d-3$ points: for each point, pick a linear form which vanishes on it but not on our final point, and multiply all of them together to find a degree $d-3$ polynomial which satisfies our conditions.
This proves our claim.

Since there's always a degree $d-1$ map available by projecting from a point on the curve, this is a sharp lower bound.
