What is the maximum number of concurrent bitangents, i.e. all intersecting at the same point, of a smooth complex projective quartic curve? Can the number of concurrent bitangents be six? 



Concurrent bitangents of a quartic curve
I think the maximum number of concurrent bitangents is at most 4. Consider the double cover $S\rightarrow \mathbb{P}^2$ branched along your quartic curve. Fix one point $q$ of $S$ above the point of intersection; each bitangent lift to a "line" in $S$ passing through $q$. The description of these lines is well-known. We can represent $S$ as the blown-up of $\mathbb{P}^2$ along 7 points $p\_1,\ldots ,p\_7 $, and assume that one of the lines is the exceptional divisor above $p\_1$. Since any pair of our "lines" in $S$ must meet, we don't have much choice: we can have the strict transform of the line in $\mathbb{P}^2$ joining $p\_1$ to another point, say $p\_2$, then the strict transform of the conic through $p\_1$ and 4 other points $\neq p\_2$, say $p\_3,\ldots ,p\_6$, and finally the the strict transform of the cubic passing through all the $p\_i$ and doubly through $p\_7$. 


So the maximum is $\leq 4$; I do not know if 4 can be actually realized. $3$ is easy, by projecting a cubic surface with 3 concurrent lines.


