Genera of Curves on a Very General Surface in P3

In this paper we consider the question of determining the geometric genera of irreducible curves lying on a very general surface $S$ of degree $d\geqslant 5$ in ${\mathbb P}^ 3$ (the cases $d\leqslant 4$ are well known). For all $d\geqslant 4$ we introduce the set ${\rm Gaps}(d)$ of all non--negative integers which are not realized as geometric genera of irreducible curves on a very general surface of degree $d$ in ${\mathbb P}^ 3$. We prove that ${\rm Gaps}(d)$ is finite and, in particular, that ${\rm Gaps}(5)= \{0,1,2\}$. The set ${\rm Gaps}(d)$ is the union of finitely many disjoint and separated integer intervals. The first of them, according to a theorem of Xu, is ${\rm Gaps}_0(d):=\left[0, \; \frac{d(d-3)}{2} - 3\right]$. We show that the next one is ${\rm Gaps}_1(d):=\left[\frac{d^2-3d+4}{2}, \; d^2 - 2d - 9\right]$ for all $d\geqslant 6$.