Weierstrass gap theorem

At every point of a smooth projective curve of genus $g$, exactly $g$ positive integers are Weierstrass gap. Indeed, each quotient $L(np)/L((n-1)p)$ has dimension zero or one, while Riemann--Roch gives $\ell (np)=n+1-g$ for all $n>2g-2$.

On a genus-two smooth projective curve, the gap set at any point is either $\{1,2\}$ or $\{1,3\}$. The integer $1$ is always a gap on a positive-genus curve. If $2$ is not a gap, then $3$ must be a gap, since pole orders are closed under addition and $2,3$ would otherwise generate every integer at least two.