Maximum number of points of intersection of a non-degenerate Hermitian variety and a cubic hypersurface
Subrata Manna
TL;DR
This work resolves the cubic-case instance of the Edoukou–Ling–Xing conjecture over ${\rm F}_{q^2}$ by proving that for $n\ge 4$ and $q\ge 7$, the maximum number of ${\rm F}_{q^2}$-rational points in common between a non-degenerate Hermitian variety ${\rm U}_n$ and a cubic hypersurface ${\mathcal C}_n$ equals $3|{\rm U}_{n-1}({\rm F}_{q^2})|-2|{\rm U}_{n-2}({\rm F}_{q^2})|$ if $n$ is even, and $(3q^2-2)|{\rm U}_{n-2}({\rm F}_{q^2})|+3$ if $n$ is odd, with parity-determined tangency conditions. The authors introduce a cubic-intersection bound sequence $B_n$ and show that exceeding these bounds forces the cubic to contain a hyperplane, enabling an explicit structural description: the maximizers are precisely unions of three hyperplanes defined over ${\rm F}_{q^2}$ that share a common ${\rm P}^{n-2}$, and whose intersection with ${\rm U}_n$ is a non-degenerate Hermitian variety; tangency vs non-tangency aligns with the parity of $n$. This algebro-geometric approach extends the understanding of extremal intersections between Hermitian varieties and low-degree hypersurfaces and provides a concrete resolution for cubic sections in the stated range of parameters. The result advances the broader conjecture by pinpointing exact extremal configurations and their geometric structure, with potential bearings on related combinatorial and coding-theoretic contexts.
Abstract
Edoukou, Ling and Xing in 2010, conjectured that in \mathbb{P}^n(\mathbb{F}_{q^2}), n \geq 3, the maximum number of common points of a non-degenerate Hermitian variety \mathcal{U}_n and a hypersurface of degree d is achieved only when the hypersurface is a union of d distinct hyperplanes meeting in a common linear space Π_{n-2} of codimension 2 such that Π_{n-2} \cap \mathcal{U}_n is a non-degenerate Hermitian variety. Furthermore, these d hyperplanes are tangent to \mathcal{U}_n if n is odd and non-tangent if n is even. In this paper, we show that the conjecture is true for d = 3 and q \geq 7.