Fix integers $r,d,s,pi$ with $rgeq 4$, $dgg s$, $r-1leq s leq 2r-4$, and $pigeq 0$. 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 $Csubset {mathbb{P}^r}$ 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.
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