On the monodromy group of everywhere tangent lines to the octic surface in 𝑃³

Abstract

Let S 0 {S_0} be an octic surface in P 3 , G = G ( 1 , 3 ) {{\mathbf {P}}^3},G = G(1,3) = Grassmannian of lines in P 3 {{\mathbf {P}}^3} , and J = { ( x , l ) | x ∈ l ∩ S 0 } ⊂ S 0 × G {\mathbf {J}} = \{ (x,l)|x \in l \cap {S_0}\} \subset {S_0} \times G . Then dim ⁡ J = 5 \dim {\mathbf {J}} = 5 . Let L = { l | l is e v e r y w h e r e tangent to S 0 } − ⊂ G {\mathbf {L}} = {\{ l|l{\text { is }}everywhere{\text { tangent to }}{S_0}\} ^ - } \subset G . Let π 2 : S 0 × G → G {\pi _2}:{S_0} \times G \to G be the projection onto the second factor. We denote its restriction to J {\mathbf {J}} also by π 2 {\pi _2} . Then the locus of everywhere tangent lines is π 2 ( L ) {\pi _2}({\mathbf {L}}) . In this article we show that the monodromy group of these lines is the full symmetric group.