Nets of conics containing a double line in $\mathrm{PG}(2,q)$, $q$ even
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911146941874176 |
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| author | Alnajjarine, Nour Lavrauw, Michel |
| author_facet | Alnajjarine, Nour Lavrauw, Michel |
| contents | This paper completes the classification of nets of conics containing at least one double line in $\mathrm{PG}(2,q)$ for $q$ even. This classification contributes to the classification of partially symmetric tensors in $\mathbb{F}_q^3 \otimes S^2 \mathbb{F}_q^3$, $q$ even. The proof is obtained using geometric and combinatorial properties of the Veronese surface in 5-dimensional projective space over the finite field of even order. In particular, the orbits of planes in $\mathrm{PG}(5,q)$ that intersect the nucleus plane of the Veronese surface in at least one point are classified. As a result, it is shown that there are exactly $18$ equivalence classes of nets in $\mathrm{PG}(2,q)$, $q$ even, containing at least one double line, $9$ of which have an empty base. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_03840 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nets of conics containing a double line in $\mathrm{PG}(2,q)$, $q$ even Alnajjarine, Nour Lavrauw, Michel Combinatorics Algebraic Geometry This paper completes the classification of nets of conics containing at least one double line in $\mathrm{PG}(2,q)$ for $q$ even. This classification contributes to the classification of partially symmetric tensors in $\mathbb{F}_q^3 \otimes S^2 \mathbb{F}_q^3$, $q$ even. The proof is obtained using geometric and combinatorial properties of the Veronese surface in 5-dimensional projective space over the finite field of even order. In particular, the orbits of planes in $\mathrm{PG}(5,q)$ that intersect the nucleus plane of the Veronese surface in at least one point are classified. As a result, it is shown that there are exactly $18$ equivalence classes of nets in $\mathrm{PG}(2,q)$, $q$ even, containing at least one double line, $9$ of which have an empty base. |
| title | Nets of conics containing a double line in $\mathrm{PG}(2,q)$, $q$ even |
| topic | Combinatorics Algebraic Geometry |
| url | https://arxiv.org/abs/2509.03840 |