Complete intersection space curves

Question

Fix two positive integers $d, e$ and assume $d>e$. Is it true that a general degree $e$ curve which lies in a complete intersection of a degree $e$ and a smooth degree $d$ surface in $\mathbb{P}^3$ is a plane curve (in the sense the radical ideal of the curve contains a linear polynomial)?

There is another way of formulating this problem. Let $H$ be the Hilbert flag scheme of pairs $(C_1, C_2)$ with $C_1 \subset C_2$, $C_2$ is a complete intersection of a degree $e$ and a smooth degree $d$ surface in $\mathbb{P}^3$ and $C_1$ is of degree $e$. The question is does there exists an irreducible component $H'$ of $H$ such that a generic element of $pr_1(H')$ is a plane curve in the sense described above (Here $pr_1$ is the natural projection map onto the first coordinate)?

In the above question we can assume $C_1$ is a local complete intersection curve in $\mathbb{P}^3$.

Answer 1

The answer is no in general. The simplest example is for $e=3$ (because a line and a conic in $\mathbb{P}^3$ are plane curves). There are plenty of twisted cubic curves lying on a smooth cubic surface in $\mathbb{P}^3$ and all lie in a complete intersection: take a general quartic passing through the twisted cubic.

If you want to be more precise, we can see a smooth cubic surface $S\subset \mathbb{P}^3$ and view it as the blow-up $\pi\colon S\to \mathbb{P}^2$ of six points in general position. An hyperplane section of $S$ corresponds, by adjunction, to the anti-canonical divisor $-K_S$, which corresponds, by ramification formula, to a cubic of $\mathbb{P}^2$ passing through the six points. Take then a line of $\mathbb{P}^2$ not passing through the six points. Its intersection with the anti-canonical divisor is $3$, hence it is a cubic curve in $\mathbb{P}^3$. Then, take a curve of degree $11$ passing through the $6$ points with multiplicity $3$, and its strict transform on $S$, added to the twisted cubic, will be the complete intersection of $S$ with a quartic. If you wonder if such curve of degree $11$ exists, just take union of lines and conics.