Quadratic hulls of osculating varieties of rational normal curves

Let $T^p{c}_r$ be the $p$-osculating variety of the rational normal curve $c_r\subset \mathbb{P}{S}^r({\mathbb{C}}^2)$. We study its quadratic hull, the scheme defined by the quadratic equations vanishing on $T^p{c}_r$. For $r\ge 2p+3$, we prove that this scheme is a complete intersection of $\left(\genfrac{}{}{0px}{}{r-2p}{2}\right)$ quadrics whenever $$p\ge \left(\genfrac{}{}{0px}{}{r-2p-1}{2}\right)\mathrm{.}$$ When the inequality is strict, the hull is integral and strictly contains $T^p{c}_r$. On the boundary its support is reducible, except for $(p,r)=(1,5)$, where the hull equals the osculating variety. The proof is local at the highest weight point. Koornwinder’s relation between Clebsch–Gordan coefficients and Hahn polynomials leads to a description of the tangent-cone equations in which each antidiagonal becomes an ordinary polynomial space of bounded degree. An incidence estimate then gives a normal integral tangent cone in the strict range, and the $S{L}_2$-action turns this local statement into global integrality.