Let $C$ be a smooth complex algebraic curve which is a complete intersection of surfaces of degree $d_1$ and $d_2$ in $\mathbb P^3$.(both $d_i >3$)
Then my question is the following : Can we always choose $n$ number of points $p_1,..p_n \in C$ for any $n$ such that the tangent lines to $C (\subset \mathbb P^3)$ at all those points are not pairwise coplanar? and If it's not true for any $n$ then can we give an upper or lower bound for such $n$?
I have an example which says for a curve which is a complete intersection of a smooth quartic and cubic surface in $\mathbb P^3$ the above statement is true for $n=2$. But I am not able to see what happens in general context.
Any help from anyone is welcome
Best Answer
This is true and has nothing to do with the curve being a complete intersection.
Let $C$ be a smooth complex space curve not contained in any hyperplane. Let $p\in C$ be a point and $l$ the tangent line at $p$. Projecting from this line, we get a morphism $\pi:C\to\mathbb{P}^1$, which is dominant since $C$ is not contained in any hyperplane and thus finite. Let $B\subset \mathbb{P}^1$ denote the union of the branch locus of $\pi$ and the images of all points in $l\cap C$. If $q\not\in \pi^{-1}(B)$, I claim that the tangent line at $q$ to $C$ doest not meet $l$ and thus they can not be coplanar. (I will leave this to you to check).
The rest is clear. Assume you have found $p_1,\ldots, p_{n-1}$ with the required property. For each $p_i$, from the previous paragraph, we only have finitely many points on $C$ whose tangent lines are coplanar with the tangent line at $p_i$. Avoiding these finitely many points for each $p_i$, we find a $p_n$ so that $p_1,\ldots, p_n$ satisfy what you need.