Refining Castelnuovo-Halphen bounds

Fix integers $r,d,s,\pi$ with $r\geq 4$, $d\gg s$, $r-1\leq s \leq 2r-4$, and $\pi\geq 0$. Refining classical results for the genus of a projective curve, we exhibit a sharp upper bound for the arithmetic genus $p_a(C)$ of an integral projective curve $C\subset {\mathbb{P}^r}$ of degree $d$, assuming that $C$ is not contained in any surface of degree $<s$, and not contained in any surface of degree $s$ with sectional genus $> \pi$. Next we discuss other types of bound for $p_a(C)$, involving conditions on the entire Hilbert polynomial of the integral surfaces on which $C$ may lie.