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

Abstract

Fix integers $r\ge 4$ and $i\ge 2$ (for $r=4$ assume $i\ge 3$). Assuming that the rational number $s$ defined by the equation $\left(\genfrac{}{}{0ex}{}{i+1}{2}\right)s+(i+1)=\left(\genfrac{}{}{0ex}{}{r+i}{i}\right)$ 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 $\le i$. 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 $\le i$. 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 ${\mathbb{P}}^{s+1}$.

4 pages