Counting maximally tangent conics relative to a cubic

Question

Is it possible to count the number of conics in $\mathbb{P}^2$ that are fully tangent at one point to a given (generic) cubic curve using basic intersection theory calculations?

The corresponding Gromov–Witten invariant (virtually counting conics relative to a cubic divisor with maximal tangency at a point) is $135/4$. After subtracting the contribution of double covers of the lines fully tangent at a point (there are 9 such lines), we get an integer that is an honest count of such conics. The latter is calculated (in much more generality) using the relative virtual localization formula but I am trying to count this case directly with basic intersection theory techniques.

Answer 1

If $X$ is a cubic and $P \in X$ is a point such that there is totally tangent at $P$ conic then $$ 6P = 2H, $$ where $H$ is the restriction to $X$ of the line class of $\mathbb{P}^2$. Thus, the set of such points is the fiber over $2H$ of the map $$ X = \mathrm{Pic}^1(X) \stackrel{6}\to \mathrm{Pic}^6(X) $$ over $2H$. But this map is a 36:1 covering, so there are 36 such points. Moreover, for each of these points the totally tangent conic is unique. Thus the answer is 36.

However, if you want to exclude double totally tangent lines, you should subtract 9; in this case the answer is 27.