The projective theorem on tangencies for dual varieties

Let $X \subset \mathbb{P}_{\mathbb{C}}^n$ be a non-degenerate, irreducible projective variety, and let $X^* \subset \mathbb{P}_{\mathbb{C}}^N$ be its projective dual. For $r \in \{0,\ldots,n-1\}$, let

$$X^*\langle r\rangle=\{H^{\bot}\in X^*: \dim\langle X_H\rangle\geq r\},$$

where $\langle\cdot\rangle$ is the scheme-theoretic linear span and $X_H$ is the tangency locus of $H$ with $X$. The projective theorem on tangencies. One has

$$\dim X^*\langle r\rangle\leq n-r-1.$$

This is the projective analogue of the Anderson–Klee result for convex bodies and expresses a dimensional constraint on the family of hyperplanes with a tangency locus whose linear span has dimension at least $r$. The source presents the assertion as a conjectural analogue because a convincing proof was not available in the earlier algebraic-geometric formulation.