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

TL;DR

This paper establishes a Castelnuovo bound for the arithmetic genus of certain projective curves not contained in low-degree hypersurfaces, proving sharpness in specific cases and analyzing extremal curves.

Contribution

It extends previous research by providing a Castelnuovo bound for the genus of projective curves under new conditions and identifies cases where the bound is sharp.

Findings

Proves sharp Castelnuovo bound for specific parameters.

Identifies conditions where the bound is not sharp and describes extremal curves.

Provides new insights into the genus of projective curves outside hypersurfaces.

Abstract

Fix integers $r\geq 4$ and $i\geq 2$. Let $C$ be a non-degenerate, reduced and irreducible complex projective curve in $\mathbb P^r$, of degree $d$, not contained in a hypersurface of degree $\leq i$. Let $p_a(C)$ be the arithmetic genus of $C$. Continuing previous research, under the assumption $d\gg \max\{r,i\}$, in the present paper we exhibit a Castelnuovo bound $G_0(r;d,i)$ for $p_a(C)$. In general, we do not know whether this bound is sharp. However, we are able to prove it is sharp when $i=2$, $r=6$ and $d\equiv 0,3,6$ (mod $9$). Moreover, when $i=2$, $r\geq 9$, $r$ is divisible by $3$, and $d\equiv 0$ (mod $r(r+3)/6$), we prove that if $G_0(r;d,i)$ is not sharp, then for the maximal value of $p_a(C)$ there are only three possibilities. The case in which $i=2$ and $r$ is not divisible by $3$ has already been examined in the literature. We give some information on the extremal curves.