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 $I_d(C_1)=I_d(C_2)$?
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$\begingroup$ What does $I_d(C_i)$ denote? The Hilbert function of the curve $C_i$? $\endgroup$– Daniel LoughranCommented Apr 15, 2012 at 13:52
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$\begingroup$ Yes. I meant the d-th graded part of the homogeneous ideal $I(C_i)$, the ideal of the curve $C_i$. $\endgroup$– Naga VenkataCommented Apr 16, 2012 at 12:49
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1 Answer
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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.
answered Dec 26, 2012 at 15:27