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Gaussian scrolls, Gaussian flags and duality

Ziv Ran

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.

Gaussian scrolls, Gaussian flags and duality

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.

Paper Structure

This paper contains 24 sections, 19 theorems, 65 equations.

Table of Contents

  1. Introduction: Gauss-deficient varieties
  2. Setup: degenerate Gauss maps as stationary scrolls
  3. Known results
  4. Contents of the paper
  5. Gaussian scrolls and flags
  6. Stationary and Gaussian scrolls: definitions
  7. Derived scroll; Gaussian flags
  8. Duality: a preview
  9. Some known constructions for stationary scrolls
  10. Joins
  11. Tangential (developable) and osculating varieties
  12. Inflation
  13. Base-change
  14. Linearity of the fibres
  15. Properties of Gauss maps and their fibres: a review
  16. ...and 9 more sections

Key Result

Theorem 1

The operation $A\mapsto A^\perp$ is a duality between the collections of Gaussian flags in $\mathbb P^N$ and in $\check \mathbb P^N$.

Theorems & Definitions (51)

  • Definition 1
  • Definition 2
  • Remark 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Remark 7
  • Theorem
  • Proposition 8
  • proof
  • ...and 41 more