Subgeometries and linear sets on a projective line
We define the splash of a subgeometry on a projective line, extending the definition of [1] to general dimension and prove that a splash is always a linear set. We also prove the converse: each linear set on a projective line is the splash of some subgeometry. Therefore an alternative description of...
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| Published in | Finite fields and their applications Vol. 34; pp. 95 - 106 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Inc
01.07.2015
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| Subjects | |
| Online Access | Get full text |
| ISSN | 1071-5797 1090-2465 |
| DOI | 10.1016/j.ffa.2015.01.006 |
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| Abstract | We define the splash of a subgeometry on a projective line, extending the definition of [1] to general dimension and prove that a splash is always a linear set. We also prove the converse: each linear set on a projective line is the splash of some subgeometry. Therefore an alternative description of linear sets on a projective line is obtained. We introduce the notion of a club of rank r, generalizing the definition from [4], and show that clubs correspond to tangent splashes. We obtain a condition for a splash to be a scattered linear set and give a characterization of clubs, or equivalently of tangent splashes. We also investigate the equivalence problem for tangent splashes and determine a necessary and sufficient condition for two tangent splashes to be (projectively) equivalent. |
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| AbstractList | We define the splash of a subgeometry on a projective line, extending the definition of [1] to general dimension and prove that a splash is always a linear set. We also prove the converse: each linear set on a projective line is the splash of some subgeometry. Therefore an alternative description of linear sets on a projective line is obtained. We introduce the notion of a club of rank r, generalizing the definition from [4], and show that clubs correspond to tangent splashes. We obtain a condition for a splash to be a scattered linear set and give a characterization of clubs, or equivalently of tangent splashes. We also investigate the equivalence problem for tangent splashes and determine a necessary and sufficient condition for two tangent splashes to be (projectively) equivalent. |
| Author | Lavrauw, Michel Zanella, Corrado |
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| Cites_doi | 10.1023/A:1005283806897 10.1007/s10623-010-9393-9 10.1016/j.disc.2009.04.007 10.1515/form.2004.029 10.1007/s10801-013-0468-3 |
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| Keywords | Subgeometry 51E20 Linear set Tangent splash Finite projective line |
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| References | Barwick, Jackson (br0020) Barwick, Jackson (br0010) Lunardon, Marino, Polverino, Trombetti (br0090) 2014; 39 Lunardon, Polverino (br0100) 2004; 16 Lavrauw, Van de Voorde (br0050) 2010; 56 Polverino (br0110) 2010; 310 Lavrauw, Van de Voorde (br0070) 2013; 20 Fancsali, Sziklai (br0040) 2006; 4 Blokhuis, Lavrauw (br0030) 2000; 81 Lidl, Niederreiter (br0080) 1997; vol. 20 Lavrauw, Van de Voorde (br0060) 2015; vol. 632 Lunardon (10.1016/j.ffa.2015.01.006_br0100) 2004; 16 Barwick (10.1016/j.ffa.2015.01.006_br0010) Polverino (10.1016/j.ffa.2015.01.006_br0110) 2010; 310 Barwick (10.1016/j.ffa.2015.01.006_br0020) Lunardon (10.1016/j.ffa.2015.01.006_br0090) 2014; 39 Fancsali (10.1016/j.ffa.2015.01.006_br0040) 2006; 4 Lavrauw (10.1016/j.ffa.2015.01.006_br0060) 2015; vol. 632 Lavrauw (10.1016/j.ffa.2015.01.006_br0070) 2013; 20 Lavrauw (10.1016/j.ffa.2015.01.006_br0050) 2010; 56 Lidl (10.1016/j.ffa.2015.01.006_br0080) 1997; vol. 20 Blokhuis (10.1016/j.ffa.2015.01.006_br0030) 2000; 81 |
| References_xml | – volume: vol. 20 year: 1997 ident: br0080 article-title: Finite Fields publication-title: Encycl. Math. Appl. – volume: 20 year: 2013 ident: br0070 article-title: Scattered linear sets and pseudoreguli publication-title: Electron. J. Comb. – ident: br0020 article-title: An investigation of the tangent splash of a subplane of – volume: vol. 632 year: 2015 ident: br0060 article-title: Field reduction and linear sets in finite geometry publication-title: Finite Fields – volume: 81 start-page: 231 year: 2000 end-page: 243 ident: br0030 article-title: Scattered spaces with respect to a spread in publication-title: Geom. Dedic. – ident: br0010 article-title: The tangent splash in – volume: 4 start-page: 89 year: 2006 end-page: 102 ident: br0040 article-title: About maximal partial 2-spreads in publication-title: Innov. Incid. Geom. – volume: 16 start-page: 663 year: 2004 end-page: 669 ident: br0100 article-title: Translation ovoids of orthogonal polar spaces publication-title: Forum Math. – volume: 39 start-page: 807 year: 2014 end-page: 831 ident: br0090 article-title: Maximum scattered linear sets of pseudoregulus type and the Segre variety publication-title: J. Algebr. Comb. – volume: 310 start-page: 3096 year: 2010 end-page: 3107 ident: br0110 article-title: Linear sets in finite projective spaces publication-title: Discrete Math. – volume: 56 start-page: 89 year: 2010 end-page: 104 ident: br0050 article-title: On linear sets on a projective line publication-title: Des. Codes Cryptogr. – volume: vol. 20 year: 1997 ident: 10.1016/j.ffa.2015.01.006_br0080 article-title: Finite Fields – volume: 81 start-page: 231 year: 2000 ident: 10.1016/j.ffa.2015.01.006_br0030 article-title: Scattered spaces with respect to a spread in PG(n,q) publication-title: Geom. Dedic. doi: 10.1023/A:1005283806897 – volume: 56 start-page: 89 year: 2010 ident: 10.1016/j.ffa.2015.01.006_br0050 article-title: On linear sets on a projective line publication-title: Des. Codes Cryptogr. doi: 10.1007/s10623-010-9393-9 – volume: 4 start-page: 89 year: 2006 ident: 10.1016/j.ffa.2015.01.006_br0040 article-title: About maximal partial 2-spreads in PG(3m−1,q) publication-title: Innov. Incid. Geom. – volume: vol. 632 year: 2015 ident: 10.1016/j.ffa.2015.01.006_br0060 article-title: Field reduction and linear sets in finite geometry – volume: 20 year: 2013 ident: 10.1016/j.ffa.2015.01.006_br0070 article-title: Scattered linear sets and pseudoreguli publication-title: Electron. J. Comb. – volume: 310 start-page: 3096 year: 2010 ident: 10.1016/j.ffa.2015.01.006_br0110 article-title: Linear sets in finite projective spaces publication-title: Discrete Math. doi: 10.1016/j.disc.2009.04.007 – ident: 10.1016/j.ffa.2015.01.006_br0010 – ident: 10.1016/j.ffa.2015.01.006_br0020 – volume: 16 start-page: 663 year: 2004 ident: 10.1016/j.ffa.2015.01.006_br0100 article-title: Translation ovoids of orthogonal polar spaces publication-title: Forum Math. doi: 10.1515/form.2004.029 – volume: 39 start-page: 807 year: 2014 ident: 10.1016/j.ffa.2015.01.006_br0090 article-title: Maximum scattered linear sets of pseudoregulus type and the Segre variety Sn,n publication-title: J. Algebr. Comb. doi: 10.1007/s10801-013-0468-3 |
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| SubjectTerms | Finite projective line Linear set Subgeometry Tangent splash |
| Title | Subgeometries and linear sets on a projective line |
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