Shadow of a cubic surface

Abstract

For a smooth cubic surface S in P^3 we can cast a shadow from a point P ∈ S that does not lie on one of the 27 lines of S onto a hyperplane H . The closure of this shadow is a smooth quartic curve. Conversely, from every smooth quartic curve we can reconstruct a smooth cubic surface whose closure of the shadow is this quartic curve. We will also present an algorithm to reconstruct the cubic surface from the bitangents of a quartic curve. The 27 lines of S together with the tangent space T_P S at P are in correspondence with the 28 bitangents or hyperflexes of the smooth quartic shadow curve. Then a short discussion on F-theory is given to relate this geometry to physics.