Fixed a genus $g$; what is the minimal integer $d$ such that for any Riemann Surface $X$ of genus $g$ there exist a positive divisor $D$ such that $l(D)\ge 2$ and $deg(D)=d$?
A Riemann-Roch argument gives that we can construct a divisor of degree $g$ with $l(D)>1$, but i don't know how to show that there exist curves of genus $g$ such that they don't have (nontrivial) meromorphic function in $l(D)$ if $deg(D)<g$,or something like that.