Edoukou, Ling and Xing in 2010, conjectured that in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${{\mathbb {P}}}^n(\mathbb {F}_{q^2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mi>n</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mi>q</mi> <mn>2</mn> </msup> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, the maximum number of common points of a non-degenerate Hermitian variety <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {U}_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">U</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> and a hypersurface of degree d is achieved only when the hypersurface is union of d distinct hyperplanes meeting in a common linear space <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Pi _{n-2}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Π</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> of codimension 2 such that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Pi _{n-2}\cap \mathcal {U}_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Π</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </msub> <mo>∩</mo> <msub> <mi mathvariant="script">U</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> is a non-degenerate Hermitian variety. Furthermore, these d hyperplanes are tangent to <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {U}_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">U</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> if n is odd and non-tangent if n is even. In this paper, we show that the conjecture is true for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$d=3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$q\ge 7$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>≥</mo> <mn>7</mn> </mrow> </math></EquationSource> </InlineEquation>.

Maximum number of points of intersection of a non-degenerate Hermitian variety and a cubic hypersurface

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发表日期：

2026年1月22日

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Subrata Manna

摘要

Edoukou, Ling and Xing in 2010, conjectured that in ${{\mathbb {P}}}^n(\mathbb {F}_{q^2})$ $P^{n} (F_{q^{2}})$ , $n\ge 3$ $n \geq 3$ , the maximum number of common points of a non-degenerate Hermitian variety $\mathcal {U}_n$ $U_{n}$ and a hypersurface of degree d is achieved only when the hypersurface is union of d distinct hyperplanes meeting in a common linear space $\Pi _{n-2}$ $Π_{n - 2}$ of codimension 2 such that $\Pi _{n-2}\cap \mathcal {U}_n$ $Π_{n - 2} \cap U_{n}$ is a non-degenerate Hermitian variety. Furthermore, these d hyperplanes are tangent to $\mathcal {U}_n$ $U_{n}$ if n is odd and non-tangent if n is even. In this paper, we show that the conjecture is true for $d=3$ $d = 3$ and $q\ge 7$ $q \geq 7$ .