Incidence and Combinatorial Properties of Linear Complexes
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2013
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| _version_ | 1866914672603561984 |
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| author | Havlicek, Hans Zanella, Corrado |
| author_facet | Havlicek, Hans Zanella, Corrado |
| contents | In this paper a generalisation of the notion of polarity is exhibited which allows to completely describe, in an incidence-geometric way, the linear complexes of $h$-subspaces. A generalised polarity is defined to be a partial map which maps $(h-1)$-subspaces to hyperplanes, satisfying suitable linearity and reciprocity properties. Generalised polarities with the null property give rise to a linear complexes and vice versa. Given that there exists for $h>1$ a linear complex of $h$-subspaces which contains no star --this seems to be an open problem over an arbitrary ground field --the combinatorial structure of a partition of the line set of the projective space into non-geometric spreads of its hyperplanes can be obtained. This line partition has an additional linearity property which turns out to be characteristic. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1304_1344 |
| institution | arXiv |
| publishDate | 2013 |
| record_format | arxiv |
| spellingShingle | Incidence and Combinatorial Properties of Linear Complexes Havlicek, Hans Zanella, Corrado Algebraic Geometry Combinatorics 51A45, 51E20, 51E23, 05B25, 05B40 In this paper a generalisation of the notion of polarity is exhibited which allows to completely describe, in an incidence-geometric way, the linear complexes of $h$-subspaces. A generalised polarity is defined to be a partial map which maps $(h-1)$-subspaces to hyperplanes, satisfying suitable linearity and reciprocity properties. Generalised polarities with the null property give rise to a linear complexes and vice versa. Given that there exists for $h>1$ a linear complex of $h$-subspaces which contains no star --this seems to be an open problem over an arbitrary ground field --the combinatorial structure of a partition of the line set of the projective space into non-geometric spreads of its hyperplanes can be obtained. This line partition has an additional linearity property which turns out to be characteristic. |
| title | Incidence and Combinatorial Properties of Linear Complexes |
| topic | Algebraic Geometry Combinatorics 51A45, 51E20, 51E23, 05B25, 05B40 |
| url | https://arxiv.org/abs/1304.1344 |