Question

Let $X \subset \mathbb{P}^3$ be a smooth degree $d$ surface containing two irreducible curves $C_1, C_2$ linearly equivalent to each other. If we assume that $X$ is general (among all degree $d$ smooth surfaces in $\mathbb{P}^3$) then is it true that $I_d(C_1)=I_d(C_2)$?

Answer 1

Let me show that the answer to this question is positive for $d>3$. Indeed, for a general surface $X$ of degree $d>3$ its Picard group is $\mathbb Z$ and is generated by $O(1)$. It follows that both curves $C_1$ and $C_2$ are complete intersections, and so they have the same Hibert function (see for example Section 13 pages 172-173 in book of Harris "first course in algebraic geometry"). Hence the statement is proved.