Classification of k-nets
A finite k-net of order n is an incidence structure consisting of k >= 3 pairwise disjoint classes of lines, each of size n, such that every point incident with two lines from distinct classes is incident with exactly one line from each of the k classes. Deleting a line class from a k-net, with k...
Elmentve itt :
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| Dokumentumtípus: | Cikk |
| Megjelent: |
2015
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| Sorozat: | EUROPEAN JOURNAL OF COMBINATORICS
48 |
| Tárgyszavak: | |
| doi: | 10.1016/j.ejc.2015.02.019 |
| mtmt: | 2906249 |
| Online Access: | http://publicatio.bibl.u-szeged.hu/29129 |
| Tartalmi kivonat: | A finite k-net of order n is an incidence structure consisting of k >= 3 pairwise disjoint classes of lines, each of size n, such that every point incident with two lines from distinct classes is incident with exactly one line from each of the k classes. Deleting a line class from a k-net, with k >= 4, gives a derived (k 1)-net of the same order. Finite k-nets embedded in a projective plane PG(2, K) coordinatized by a field K of characteristic 0 only exist for k = 3, 4, see Korchmaros et al. (2014). In this paper, we investigate 3-nets embedded in PG(2, K) whose line classes are in perspective position with an axis r, that is, every point on the line r incident with a line of the net is incident with exactly one line from each class. The problem of determining all such 3-nets remains open whereas we obtain a complete classification for those coordinatizable by a group. As a corollary, the (unique) 4-net of order 3 embedded in PG(2, K) turns out to be the only 4-net embedded in PG(2, K) with a derived 3-net which can be coordinatized by a group. Our results hold true in positive characteristic under the hypothesis that the order of the k-net considered is smaller than the characteristic of K. (C) 2015 Elsevier Ltd. All rights reserved. |
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| Terjedelem/Fizikai jellemzők: | 177-185 |
| ISSN: | 0195-6698 |