Cr-invariants for surfaces in 4-space

TL;DR

This paper introduces cross-ratio invariants for surfaces in four-dimensional space, extending previous work in three dimensions, and uses them to analyze singularities and asymptotic configurations of the surfaces.

Contribution

It develops new cross-ratio invariants for 4-space surfaces and applies them to classify singularities and stable asymptotic configurations.

Findings

Cross-ratio invariants characterize surface singularities in 4-space.

Two moduli in the 4-jet of the surface are recovered using these invariants.

Stable configurations of asymptotic curves are identified.

Abstract

We establish cross-ratio invariants for surfaces in 4-space in an analogous way to Uribe-Vargas's work for surfaces in 3-space. We study the geometric locii of local and multi-local singularities of ortogonal projections of the surface. The cross-ratio invariants at $P_3(c)$-points are used to recover two moduli in the 4-jet of the projective parametrization of the surface and show the stable configurations of asymptotic curves.