Castelnuovo bound says that if we have a function field(algebraic curve) $F$ and a divisor on it $D$ then: $g\leq c\frac{\deg(D)^2}{\ell(D)}$(where $c$ is some global constant say 2 and $g$ is a genus of the curve). I would like to ask if the converse is true? My question is if the converse is true for every $\ell(D)$? Formally the question is the following:

Does there exists a constant $c$ such that for every function field $F$ and for every integer $2\leq l \leq g$ there exists a divisor $D$ with $\ell(D)= l$ and $g\geq c\frac{\deg(D)^2}{\ell(D)}$?

`$\deg(D) \le \sqrt{g} \Longrightarrow \ell(D) \le 1$`

is wrong in general: for a hyperelliptic curve, with $D$ twice a Weierstrass point, we have $\ell(D) = 2$ and $\deg(D) = 2$, which is $\le \sqrt{g}$ for $g$ sufficiently large. – Michael Stoll Apr 29 '12 at 16:43