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 $I_d(C_1)=I_d(C_2)$?

$\begingroup$ What does $I_d(C_i)$ denote? The Hilbert function of the curve $C_i$? $\endgroup$– Daniel LoughranApr 15, 2012 at 13:52

$\begingroup$ Yes. I meant the dth graded part of the homogeneous ideal $I(C_i)$, the ideal of the curve $C_i$. $\endgroup$– Naga VenkataApr 16, 2012 at 12:49
1 Answer
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 172173 in book of Harris "first course in algebraic geometry"). Hence the statement is proved.