Osculating properties of decomposable scrolls

TL;DR

This paper investigates the osculating spaces and inflectional loci of decomposable scrolls, extending known ideas to broader classes and introducing new uninflected surface scrolls with detailed geometric properties.

Contribution

It generalizes the study of osculating properties from rational normal scrolls to decomposable scrolls of any genus, and introduces a new class of uninflected surface scrolls.

Findings

Relation between inflectional loci of scrolls and their generating curves

Identification and characterization of a new class of uninflected surface scrolls

Analysis of additional components of the second discriminant locus

Abstract

Osculating spaces of decomposable scrolls (of any genus and not necessarily normal)are studied and their inflectional loci are related to those of their generating curves by using systematically an idea introduced by Piene and Sacchiero in the setting of rational normal scrolls. In this broader setting the extra components of the second discriminant locus - deriving from flexes - are investigated and a new class of uninflected surface scrolls is presented and characterized. Further properties related to osculation are discussed for (not necessarily decomposable) scrolls.