So this question is directly related to a comment made by David Mumford in his 
Lecture 1 given at U. Michigan in 1974 entitled: *What is a curve and how explicitly can we describe them ?* 


Mumford claims that if you take a (non-hyper elliptic) smooth projective curve $C$ over $\mathbb{C}$ of genus 3 and embed it in $\mathbf{P}^2$ via its canonical map (denoting the image of the curve again by $C$), then there are exactly $108$ points $x\in C$ for which there is a conic passing through $x$ with contact order (with respect to $C$) equal to $6$.


Q1: How does one prove that you have only finitely many such conics touching $C$
and have contact order $6$?


This seems to suggest, that for most points $P\in C$, the best contact order of 
a conic passing through $P$ is $5$.


Q2: In general if $C\subseteq \mathbf{P}^2$ is a fixed embedded smooth projective curve and $x\in C$ is a point, then for a fixed degree $d$, how does one compute the maximum contact order at $x$ among all smooth projective curves $D$ of degree $d$ in $\mathbf{P}^2$ passing through $x$ (is this computable)? 



Higher Weierstrass points on curves of genus 3
Will is right of course: there exists a curve of degree $d$ with a contact of order $k$ with $C$ at $P$ iff $h^0(C,\mathcal{O}\_C(dH-kP))>0$, where $H$ is the divisor of a line. So:


Q1) $2H-6P$ has degree 2, so the condition is that it lies on the canonical theta divisor $\Theta $ of $J^2C$.
So we are looking at the intersection of $\Theta $ with $6\_\*C$, the image of $C$ under multiplication by 6 in $JC$ (I am implicitely choosing some base point here to identify $J^2C$ with $JC$). Now $6\_\*$ acts as $6^p$ on $H\_p(JC)$, so the (co)homology class of $6\_\*C$ is $6^2[C]=6^2\frac{\Theta ^2}{2} $. Thus $(\Theta .6\_\*C)=6^2\frac{\Theta ^3}{2}=6^2.3=108$.


Q2) The same ideas will give you the general answer. Let $e:=\deg(C)$, $g:=g(C)$. We look at $dH-kP$. 
 If $k<de-(g-1)$, there are always (many) curves of degree $d$ with contact of order $k$ at $P$. If $k>de-(g-1)$, there is none in general; there can be some, of course, for some particular pairs $(C,x)$. The interesting case is $k=de-(g-1)$; then we find $N$ contact points, with $N=(\Theta . k\_\*C)=k^2g$ (of course these points must be counted with multiplicity). 


