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 $RP^3$](https://mathoverflow.net/questions/58000/polynomial-contact-structures-on-rp3)
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.



Thom polynomial for contact algebraic structures
Let $i:C \to \mathbb P^3$ be the normalization of an irreducible curve $C\_0\subset \mathbb P^3$ of degree $d$ and geometric genus $g$. 


If $\mathcal D$ is a distribution on $\mathbb P^3$ of degree $p$ then it is defined by a section $\omega$ of $\Omega^1\_{\mathbb P^3} \otimes \mathcal O\_{\mathbb P^3}(p+2)$. To compute the tangencies between $C\_0$ and $\mathcal D$ we pull-back $\omega$ to $C$ using $i$. What we get is a section of $\Omega^1\_C \otimes \mathcal O\_C(p+2)$. Notice that $\mathcal O\_C(p+2) = i^\* \mathcal O\_{\mathbb P^3}(p+2)$ is a line-bundle of degree $d(p+2)$ over $C$.


If $i^\* \omega$ vanishes identically then $C\_0$ is everywhere tangent to $\mathcal D$. Otherwise, it is a section of a line-bundle of degree equal to $\deg(\Omega^1\_C) + \deg(\mathcal O\_C(p+2)$. As such it has exactly $2g -2 + d(p+2)$ zeros counted with multiplicities. Therefore
$$
f(d,g,p) = 2g -2 + d(p+2) +1 
$$ 
is a polynomial. If one recalls that the genus of degree $d$ irreducible curve is bounded by $(d-1)(d-2)/2$ then one sees that
$$
f(d,g,p) \le d( p + d -1) + 1 . 
$$
This is in accordance with Serge R. claim that the bound can be taken independent of $g$. 


The argument above works equally well for integrable and non-integrable distributions, and also works for codimension one distributions on $\mathbb P^n$, $n\ge 2$. 


