Curves on Rational Surfaces with Hyperelliptic Hyperplane Sections

TL;DR

This paper investigates the existence of certain algebraic curves on rational surfaces with hyperelliptic hyperplane sections, establishing non-existence gaps in specific domains and proposing a conjecture about the structure of these domains.

Contribution

It introduces new constructions of curves on rational surfaces and proves the absence of gaps in a defined domain, advancing understanding of the Halphen-Castelnuovo problem.

Findings

No gaps in the domain D1,n for the existence of curves.

Constructed curves on rational surfaces with hyperelliptic hyperplane sections.

Conjecture that D2,n is the lacunary domain for such curves.

Abstract

In this article we study, given a pair of integers (d,g), the problem of existence of a smooth, irreducible, non-degenerate curve in the projective n-domensional space of degree d and genus g (the Halphen-Castelnuovo Problem). We define two domains from the (d,g)-plane, D1,n and D2,n, and we prove that there is no gap in D1,n. This follows by constructing curves on some rational surfaces with hyperelliptic hyperplane sections, and from some previous Theorems of Ciliberto, Sernesi, and of the author. Moreover, in the last section, based on some results of Horrowitz, Ciliberto, Harris, Eisenbud, we Conjecture that D2,n is the right lacunary domain.