Genera of curves on a very general surface in $P^3$

In this paper we consider the question of determining the geometric genera of irreducible curves lying on a very general surface $S$ of degree $d$ at least 5 in ${\mathbb{P}}^3$ (the cases $d⩽4$ are well known). We introduce the set $Gaps(d)$ of all non-negative integers which are not realized as geometric genera of irreducible curves on $S$. We prove that $Gaps(d)$ is finite and, in particular, that $Gaps(5)=\{0,1,2\}$. The set $Gaps(d)$ is the union of finitely many disjoint and separated integer intervals. The first of them, according to a theorem of Xu, is $Gaps_0(d):=[0, \frac{d(d-3)}{2} - 3]$\mathrm{.}Weshowthatthenextoneis$Gaps_1(d):= [\frac{d^2-3d+4}{2}, d^2-2d-9]$forall$d \geqslant 6$.