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Let $X$ be a smooth degree $d$ surface in $\mathbb{P}^3$ and $C$ be a Cartier divisor on $X$. Assume that the degree of $C$ is less than $d$. We know that if $C$ is reduced then $H^1(\mathcal{O}_X(-C)(d))=0$. This follows from Castelnuovo-Mumford regularity. The question is how much can we extend this result to arbitrary Cartier divisor. In other words, is it true that for any Cartier divisor of degree less than $d$, $H^1(\mathcal{O}_X(-C)(d))=0$? If this is not true in such a generality, can we find a bound on the degree of the Cartier divisor for which this will be true?

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