Classification of smooth congruences with a fundamental curve

TL;DR

This paper classifies smooth varieties of lines in projective space where all lines meet a curve, providing a comprehensive understanding of such structures and their relation to scrolls over a curve.

Contribution

It offers a complete classification and construction method for smooth congruences of lines in projective space meeting a curve, linking to scrolls over a curve.

Findings

Classified all smooth congruences of lines meeting a curve

Constructed explicit examples of these varieties

Connected the classification to scrolls over a curve

Abstract

We give a classification and a construction of all smooth $(n-1)$-dimensional varieties of lines in ${\bf P}\sp n$ verifying that all their lines meet a curve. This also gives a complete classification of $(n-1)$-scrolls over a curve contained in $G(1,n)$.