On the cuspidal locus in the dual varieties of Segre quartic surface

TL;DR

This paper studies the space of hyperplane sections with cusps on Segre quartic surfaces, revealing conditions for emptiness or connectedness and describing the geometric structure of their closures.

Contribution

It characterizes the cuspidal hyperplane sections of Segre quartic surfaces, identifying when this space is empty or connected and describing its birational geometry.

Findings

Space is empty for two Segre surface types.

Space is a connected surface for other types.

Closure is birational to the surface or a double cover with specific branch lines.

Abstract

Motivated by a kind of Penrose correspondence, we investigate the space of hyperplane sections of Segre quartic surfaces which have an ordinary cusp. We show that the space of such hyperplane sections is empty for two kinds of Segre surfaces, and it is a connected surface for all other kinds of Segre surfaces. We also show that when it is non-empty, the closure of the space is either birational to the surface itself or birational to a double covering of the surface, whose branch divisor consists of some specific lines on the surface.