Let $X$ be a smooth plane projective quintic curve (over $\mathbb C$). Then we know that it has gonality $4$. Assume that it has genus $g(X)=6$. Then my question is the following:



> 
> Is it necessarily true that for every line bundle $A$ on $X$ with
> $h^0(A) \geq 2$ one has $\text{deg}(A)\geq h^0(A)+2$?
> 
> 
> 


Gonality $4$ means minimum degree of line bundles with atleast $2$ sections is $4$. On the other hand we have for any line bundle with atleast $2$ sections and with $h^1(A) \geq 2$, $\text{deg}(A) \geq 2h^0(A)-2$. But then it's not quite clear to me that how these two facts on gonality and clifford index (and may be Riemann-Roch) give us an affirmative answer to the question. Maybe I'm missing something obvious. Does there exist a more direct proof in the literature?



On degree and section of a line bundle on a smooth plane quintic
This is true, and can be shown by an induction argument on $h^0(A)$.
If $h^0(A)=2$, then $\deg(A)\geq 4$ since the gonality of $X$ is $4$.
If $h^0(A)>2$, let $p\in X$ be a point in the support of an effective divisor representing $A$. Since $\mathrm{H}^0(X,A(-p))\subset \mathrm{H}^0(X,A)$ is a subspace of codimension $\leq 1$, we obtain the inequality $$ h^0(A(-p))\geq h^0(A)-1 .$$
This implies that $h^0(A(-p))\geq 2$, so using the induction hypothesis and the inequality again we obtain $$\deg(A)-1 = \deg(A(-p)) \geq h^0(A(-p))+2\geq h^0(A)+1 .$$
Add $1$ to both sides.


Remark: this argument only uses that the gonality of $X$ is $4$.


