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I'm having a problem computing something that seems easy, at least from classical algebraic geomtery point of view. Take a complex smooth surface $S$ defined over $\mathbb{C}$ in $\mathbb{P}^3$, and choose a generic point on it $p \in S$. What is the number of bitangents to $S$ passing through $p$? I guess one could compute it with incidence structures, but I could not find how. Thanks!

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