Curves contracted by the Gauss map

TL;DR

This paper characterizes smooth curves contracted by the Gauss map in singular projective varieties using normal bundles and reveals deformation obstructions and non-reduced components in the Hilbert scheme.

Contribution

It provides a new characterization of contracted smooth curves via normal bundles and analyzes their deformation properties in normal varieties.

Findings

Contracted lines have local obstructions for embedded deformation.

Each component of the Hilbert scheme containing such lines is non-reduced.

Normality of the variety influences the deformation behavior of contracted lines.

Abstract

Given a singular projective variety in some projective space, we characterize the smooth curves contracted by the Gauss map in terms of normal bundles. As a consequence, we show that if the variety is normal, then a contracted line always has local obstruction for the embedded deformation and each component of the Hilbert scheme where the line lies is non-reduced everywhere.