Tetrads of lines spanning PG(7,2)
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arXiv
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| Format: | Preprint |
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2012
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| _version_ | 1866914675580469248 |
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| author | Shaw, Ron Gordon, Neil Havlicek, Hans |
| author_facet | Shaw, Ron Gordon, Neil Havlicek, Hans |
| contents | Our starting point is a very simple one, namely that of a set L_4 of four mutually skew lines in PG(7,2): Under the natural action of the stabilizer group G(L_4) < GL(8,2) the 255 points of PG(7,2) fall into four orbits omega_1, omega_2, omega_3 omega_4; of respective lengths 12, 54, 108, 81: We show that the 135 points in omega_2 \cup omega_4 are the internal points of a hyperbolic quadric H_7 determined by L_4; and that the 81-set omega_4 (which is shown to have a sextic equation) is an orbit of a normal subgroup G_81 isomorphic to (Z_3)^4 of G(L_4): There are 40 subgroups (isomorphic to (Z_3)^3) of G_81; and each such subgroup H < G_81 gives rise to a decomposition of omega_4 into a triplet of 27-sets. We show in particular that the constituents of precisely 8 of these 40 triplets are Segre varieties S_3(2) in PG(7,2): This ties in with the recent finding that each Segre S = S_3(2) in PG(7,2) determines a distinguished Z_3 subgroup of GL(8,2) which generates two sibling copies S'; S" of S. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1202_5745 |
| institution | arXiv |
| publishDate | 2012 |
| record_format | arxiv |
| spellingShingle | Tetrads of lines spanning PG(7,2) Shaw, Ron Gordon, Neil Havlicek, Hans Combinatorics Algebraic Geometry 51E20, 05B25, 15A69 Our starting point is a very simple one, namely that of a set L_4 of four mutually skew lines in PG(7,2): Under the natural action of the stabilizer group G(L_4) < GL(8,2) the 255 points of PG(7,2) fall into four orbits omega_1, omega_2, omega_3 omega_4; of respective lengths 12, 54, 108, 81: We show that the 135 points in omega_2 \cup omega_4 are the internal points of a hyperbolic quadric H_7 determined by L_4; and that the 81-set omega_4 (which is shown to have a sextic equation) is an orbit of a normal subgroup G_81 isomorphic to (Z_3)^4 of G(L_4): There are 40 subgroups (isomorphic to (Z_3)^3) of G_81; and each such subgroup H < G_81 gives rise to a decomposition of omega_4 into a triplet of 27-sets. We show in particular that the constituents of precisely 8 of these 40 triplets are Segre varieties S_3(2) in PG(7,2): This ties in with the recent finding that each Segre S = S_3(2) in PG(7,2) determines a distinguished Z_3 subgroup of GL(8,2) which generates two sibling copies S'; S" of S. |
| title | Tetrads of lines spanning PG(7,2) |
| topic | Combinatorics Algebraic Geometry 51E20, 05B25, 15A69 |
| url | https://arxiv.org/abs/1202.5745 |