On the genus of projective curves not contained in hypersurfaces of given degree

TL;DR

This paper establishes an upper bound for the genus of certain complex projective curves not contained in low-degree hypersurfaces, linking it to Castelnuovo's bound and characterizing extremal cases via rational normal scroll surfaces.

Contribution

It introduces a new genus bound for projective curves based on their degree and containment properties, extending Castelnuovo's classical results and characterizing extremal curves through the existence of specific surfaces.

Findings

The genus bound coincides with Castelnuovo's bound in certain projective spaces.

Extremal curves are projections of Castelnuovo's curves when specific surfaces exist.

The bound is sharp for i=2 and i=3, with known existence of the associated surfaces.

Abstract

Fix integers $r\geq 4$ and $i\geq 2$ (for $r=4$ assume $i\geq 3$). Assuming that the rational number $s$ defined by the equation $\binom{i+1}{2}s+(i+1)=\binom{r+i}{i}$ is an integer, we prove an upper bound for the genus of a reduced and irreducible complex projective curve in $\mathbb P^r$, of degree $d\gg s$, not contained in hypersurfaces of degree $\leq i$. It turns out that this bound coincides with the Castelnuovo's bound for a curve of degree $d$ in $\mathbb P^{s+1}$. We prove that the bound is sharp if and only if there exists an integral surface $S\subset \mathbb P^r$ of degree $s$, not contained in hypersurfaces of degree $\leq i$. Such a surface, if existing, is necessarily the isomorphic projection of a rational normal scroll surface of degree $s$ in $\mathbb P^{s+1}$. The existence of such a surface $S$ is known for $i=2$ and $i=3$. It follows that, when $i=2$ or $i=3$, the bound is sharp, and the extremal curves are isomorphic projection in $\mathbb P^r$ of Castelnuovo's curves of degree $d$ in $\mathbb P^{s+1}$.