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Let $S$ be a smooth quartic surface in $\mathbb{P}^3$, given by an equation $F=0$. Through a general point $p$ of $S$ pass two tritangents, that is, lines with a contact of order 3 at $p$. This defines a two-sheeted covering of a Zariski open subset of $S$; I want this covering to split, that is, I want $S$ to have two distinct families of tritangents. By a result of Predonzan and B. Segre (1960), this is equivalent to say that the Hessian surface $H$ of $F$ (defined by $\ \det F''=0$) meets $S$ with even multiplicity, i.e. $H_{|S}=2C$ for a curve $C$ in $S$.

Predonzan gives many examples of singular such surfaces, but for smooth ones I know only the Fermat surface. Any other?

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