Rational Quartic Symmetroids

Abstract

The study of generic quartic symmetroids in projective 3-space dates back to Cayley, but little is known about the special specimens. This thesis sets out to survey the possibilities in the non-generic case. We prove that the Steiner surface is a symmetroid. We also find quartic symmetroids that are double along a line and have two to eight isolated nodes, symmetroids that are double along two lines and have zero to four isolated nodes and symmetroids that are double along a smooth conic section and have two to four isolated nodes. In addition, we present degenerated symmetroids with fewer isolated singularities. A symmetroid that is singular along a smooth conic section and a line is given. This culminates in a 21-dimensional family of rational quartic symmetroids.