Maximum number of points of intersection of a non-degenerate Hermitian variety and a cubic hypersurface
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914084746690560 |
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| author | Manna, Subrata |
| author_facet | Manna, Subrata |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_13106 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Maximum number of points of intersection of a non-degenerate Hermitian variety and a cubic hypersurface Manna, Subrata Algebraic Geometry 4G05, 14G15, 05B25 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. |
| title | Maximum number of points of intersection of a non-degenerate Hermitian variety and a cubic hypersurface |
| topic | Algebraic Geometry 4G05, 14G15, 05B25 |
| url | https://arxiv.org/abs/2504.13106 |