On varieties with higher osculating defect

TL;DR

This paper generalizes Terracini's results on varieties with tangent defect by characterizing higher osculating defect varieties through Jacobians of fundamental forms and their containment in scrolls.

Contribution

It introduces a new characterization of varieties with higher order osculating defect using Jacobians and fundamental forms, extending previous results.

Findings

Varieties with higher osculating defect are characterized via Jacobians of higher fundamental forms.

Varieties with small higher fundamental forms are contained in scrolls.

The method of moving frames is used to achieve these generalizations.

Abstract

In this paper, using the method of moving frames, we generalise some of Terracini's results on varieties with tangent defect. In particular, we characterise varieties with higher order osculating defect in terms of Jacobians of higher fundamental forms and moreover we characterise varieties with "small" higher fundamental forms as contained in scrolls.