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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)?

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  • $\begingroup$ Isn't this $\max d: \dim H^0(C,\mathcal O_C(n-dP)) >0$? $\endgroup$
    – Will Sawin
    Commented Feb 6, 2014 at 5:35

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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).

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    $\begingroup$ This is a great question/answer. Note further that the number $108$ includes the cases where the "sextactic" conic is singular. It turns out that the points at which the sextactic conic is singular are none other than the flexes of $C$ -- indeed, the doubled tangent line at a flex has contact of order $6$ with $C$ at the flex. There are precisely $3d(d-2) = 24$ such flexes on a genus $3$ plane curve, so the number of points at which the sextactic conic is smooth is only $84$. $\endgroup$
    – Ashvin Swaminathan
    Commented Mar 16, 2017 at 23:16
  • $\begingroup$ Just two small comments. Those 108 points are exactly the 2-Weierstrass points of the quartic (counted with multiplicities), and the 84 counted with multiplicities, are the 2-Weierstrass which are not Weierstrass. Moreover, I think that the condition $h^0(C,dH-kp))>0$ is equivalent to the existence of a curve of degree š¯‘‘ with a contact of order $at$ $least$ $k$ with $C$ at $p$. Indeed, it can have contact 7 or 8. $\endgroup$
    – Lidia
    Commented Jan 29, 2023 at 17:54

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