Characterising ovoidal cones by their intersection numbers
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866914691712811008 |
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| author | De Bruyn, Bart Van de Voorde, Geertrui |
| author_facet | De Bruyn, Bart Van de Voorde, Geertrui |
| contents | In this paper, we characterise ovoidal cones by their intersection numbers. We first show that a set of points of $\mathrm{PG}(4,q)$ which intersects planes in $1$, $q+1$ or $2q+1$ points is either an ovoidal cone or a parabolic quadric, unless $q=3$, in which case also a sporadic example with automorphism group $M_{11}$ exists. We then show that a set of points of $\mathrm{PG}(4,q)$ which blocks all planes and intersects solids in $q+1$, $q^2+1$ or $q^2+q+1$ points is a plane or an ovoidal cone, and determine all examples that arise when the blocking condition is omitted. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_05666 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Characterising ovoidal cones by their intersection numbers De Bruyn, Bart Van de Voorde, Geertrui Combinatorics 51E20, 51A50 In this paper, we characterise ovoidal cones by their intersection numbers. We first show that a set of points of $\mathrm{PG}(4,q)$ which intersects planes in $1$, $q+1$ or $2q+1$ points is either an ovoidal cone or a parabolic quadric, unless $q=3$, in which case also a sporadic example with automorphism group $M_{11}$ exists. We then show that a set of points of $\mathrm{PG}(4,q)$ which blocks all planes and intersects solids in $q+1$, $q^2+1$ or $q^2+q+1$ points is a plane or an ovoidal cone, and determine all examples that arise when the blocking condition is omitted. |
| title | Characterising ovoidal cones by their intersection numbers |
| topic | Combinatorics 51E20, 51A50 |
| url | https://arxiv.org/abs/2402.05666 |