Gaussian scrolls, Gaussian flags and duality

TL;DR

This paper explores the properties and dualities of Gaussian scrolls and flags in projective varieties, revealing new relationships and structures within algebraic geometry.

Contribution

It introduces a duality for Gaussian scrolls and flags, and studies their geometric and duality properties in projective varieties.

Findings

Gaussian scrolls are dual to derived or tangent developable scrolls in dual space

Gaussian scrolls are the 'leading edge' of their derived stationary scrolls

The study establishes a duality framework for Gaussian flags and scrolls

Abstract

A projective variety whose Gauss map has positive dimensional fibres corresponds to a special kind of scroll called \emph{Gaussian}. A Gaussian scroll is a member of a canonical derived \emph{ Gaussian flag}. We introduce a duality in the class of Gaussian scrolls and flags and study its consequences. In particular, a Gaussian scroll is dual to the derived or tangent developable scroll of a Gaussian scroll in the dual projective space, and is the 'leading edge' or antiderived scroll of its derived stationary scroll.