On the monodromy group of everywhere tangent lines to the octic surface in $\textbf {P}^ 3$

Abstract:

Let ${S_0}$ be an octic surface in ${{\mathbf {P}}^3},G = G(1,3)$ = Grassmannian of lines in ${{\mathbf {P}}^3}$, and ${\mathbf {J}} = \{ (x,l)|x \in l \cap {S_0}\} \subset {S_0} \times G$. Then $\dim {\mathbf {J}} = 5$. Let ${\mathbf {L}} = {\{ l|l{\text { is }}everywhere{\text { tangent to }}{S_0}\} ^ - } \subset G$. Let ${\pi _2}:{S_0} \times G \to G$ be the projection onto the second factor. We denote its restriction to ${\mathbf {J}}$ also by ${\pi _2}$. Then the locus of everywhere tangent lines is ${\pi _2}({\mathbf {L}})$. In this article we show that the monodromy group of these lines is the full symmetric group.