How many planes are totally tangent to a curve of $\mathbb{P}^3$ which is intersection of a generic quadric and a generic cubic?

Or equivalently, considering the cubic as a blow up of $\mathbb{P}^2$ in $P_1,...,P_6$, how many cubic curves passing through $P_1,...,P_6$ in $\mathbb{P}^2$ are totally tangent to a fixed generic sextic curve with simple nodes at $P_1,...,P_6$?

  • $\begingroup$ Can you remind me what totally tangent means? $\endgroup$ Sep 25 '13 at 13:57
  • $\begingroup$ Sorry, I meant exactly what Felipe wrote: a totally tangent plane is a plane tangent at every point of its intersection with the curve. $\endgroup$
    – sqrt2sqrt2
    Sep 26 '13 at 14:43

Totally tangent is not standard terminology and I am not sure what it means. Taking a guess that it means that the plane is tangent at every point it meets the curve, it means that the divisor cut by the plane is of the form $2D$ for a positive divisor $D$ of degree $3$. An intersection of a cubic and a quadric is a canonically embedded curve of genus $4$ so $2D$ is a canonical divisor and $D$ is a theta characteristic (there are $2^8$ of them). As $D$ is effective and the curve is generic, you want an odd theta characteristic, so $2^3(2^4-1)=120$ of those. See Mumford, Ann Sci ENS 1971.


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