# Scheme-theoretic Whitney conditions and applications to tangency of projective varieties

## TL;DR

This paper introduces a scheme-theoretic version of Whitney condition a, linking it to the smoothness of the dual variety, and applies it to problems in tangency, osculating planes, and algebraic boundaries in projective and real algebraic geometry.

## Contribution

It develops a scheme-theoretic framework for Whitney conditions and demonstrates its applications to tangency and boundary problems in algebraic geometry.

## Key findings

- Proves a Bertini-type theorem for osculating planes of space curves.
- Establishes a connection between Whitney conditions and dual variety smoothness.
- Generalizes a theorem on algebraic boundaries of real varieties.

## Abstract

We investigate a scheme-theoretic variant of Whitney condition a. If X is a projec-tive variety over the field of complex numbers and Y $\subset$ X a subvariety, then X satisfies generically the scheme-theoretic Whitney condition a along Y provided that the pro-jective dual of X is smooth. We give applications to tangency of projective varieties over C and to convex real algebraic geometry. In particular, we prove a Bertini-type theorem for osculating plane of smooth complex space curves and a generalization of a Theorem of Ranestad and Sturmfels describing the algebraic boundary of an affine compact real variety.