Genera of curves on a very general surface in P³

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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 P³ (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 Gaps₀(d):=[0, d(d-3)/2 - 3]. We show that the next one is Gaps₁(d):= [d²-3d+4/2, d²-2d-9] for all d 6.