Singularities of Tangent Varieties to Curves and Surfaces

TL;DR

This paper classifies generic singularities of tangent varieties to curves and surfaces in projective spaces, using geometric structures and differential systems, and provides new insights into their diffeomorphism types.

Contribution

It offers a comprehensive diffeomorphism classification of tangent variety singularities for curves and surfaces, extending existing theories with new geometric and algebraic techniques.

Findings

Classified generic singularities of tangent varieties to curves in projective space.

Classified singularities of tangent varieties to contact-integral curves.

Provided foundational results on tangent varieties to surfaces and Legendre surfaces.

Abstract

It is given the diffeomorphism classification on generic singularities of tangent varieties to curves with arbitrary codimension in a projective space. The generic classifications are performed in terms of certain geometric structures and differential systems on flag manifolds, via several techniques in differentiable algebra. It is provided also the generic diffeomorphism classification of singularities on tangent varieties to contact-integral curves in the standard contact projective space. Moreover we give basic results on the classification of singularities of tangent varieties to generic surfaces and Legendre surfaces.