Genera of curves on a very general surface in IP^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 P^3 (the cases d at most 4 are well known). For all d at least 5 we introduce the set 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 P^3. 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, (1/2) d(d−3)− 3]. We show that the next one is Gaps_1(d) := [(1/2) (d^2−3d+4), d^2 − 2d − 9] for all d at least 6.