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

TL;DR

This paper investigates the possible geometric genera of irreducible curves on very general surfaces in projective three-space, establishing finiteness of non-realizable genera and explicitly describing the structure of these gaps.

Contribution

It introduces the set Gaps(d) of non-realizable genera on a very general surface of degree d in P^3 and characterizes its structure, including explicit bounds for the gaps.

Findings

Gaps(d) is finite for all d ≥ 5.

Gaps(d) consists of finitely many disjoint integer intervals.

Explicit descriptions of the first two gap intervals for d ≥ 5.

Abstract

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 $d \leqslant 4$ 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 $Gaps_0(d):=[0, \frac{d(d-3)}{2} - 3]$. We show that the next one is $Gaps_1(d):= [\frac{d^2-3d+4}{2}, d^2-2d-9]$ for all $d \geqslant 6$.