Quadratic hulls of osculating varieties of rational normal curves
Abstract
Let Tp cr be the p-osculating variety of the rational normal curve cr⊂ℙSr(ℂ2). We study its quadratic hull, the scheme defined by the quadratic equations vanishing on Tp cr. For r≥2p+3, we prove that this scheme is a complete intersection of binomr−2p2 quadrics whenever p≥binomr−2p−12. When the inequality is strict, the hull is integral and strictly contains Tp cr. 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 SL2-action turns this local statement into global integrality.
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BibTeX
@misc{massri2026quadratic,
title = {{Quadratic hulls of osculating varieties of rational normal curves}},
author = {César Massri},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28499},
eprint = {2610.28499},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28499},
}
RIS
TY - UNPB
TI - Quadratic hulls of osculating varieties of rational normal curves
AU - César Massri
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28499
UR - https://arxiv.org/abs/2610.28499
AB - Let T^p c_r be the p-osculating variety of the rational normal curve c_r⊂ℙS^r(ℂ^2). We study its quadratic hull, the scheme defined by the quadratic equations vanishing on T^p c_r. For r≥2p+3, we prove that this scheme is a complete intersection of binomr−2p2 quadrics whenever p≥binomr−2p−12. 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 SL_2-action turns this local statement into global integrality.
ER -