Let's consider a algebraic contact structure $P$ on $\mathbb CP^3$ and a algebraic curve $C$ degree $d$ and genus $g$. Let's assume that contact structure has degree $p$ (see Polynomial contact structures on $RP^3$ about algebraic contact structure).
It seems that there is some constant $f(d,g,p)$ (maybe, even polynomial!) such that if $C$ is tangent to $P$ at $f$ points then $C$ is tangent to $P$ everywhere.
For example, it is easy to prove that $f(d,0,0)$ equals $2d-1$.
Somebody can expect that this question is about some homological conditions : generic curve is tangent to $P$ at $a$ points, so, if it is tangent to $P$ at $a+1$ points then it is tangent everywhere. It seems that it is true because pull-back of contact form to tangent bundle of $C$ is a holomorphic form, so it has some degree..
So, my questions are:
1) How can we prove that $f(d,g,p)$ exists? At least for some values $d,g,p$? It seems that the case $p=0$ is the most easy.
2) Is it true that $f$ is a polynomial? (Thom polynomial of something...) I'm sure that it is known or similar constructions are already examined.