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Absolute points of correlations of $$PG(3,q^n)$$ P G ( 3 , q n )
Journal of Algebraic Combinatorics ( IF 0.6 ) Pub Date : 2020-08-28 , DOI: 10.1007/s10801-020-00970-3
Giorgio Donati , Nicola Durante

The sets of the absolute points of (possibly degenerate) polarities of a projective space are well known. The sets of the absolute points of (possibly degenerate) correlations, different from polarities, of \({{\mathrm{PG}}}(2,q^n)\), have been completely determined by B.C. Kestenband in 11 papers from 1990 to 2014, for non-degenerate correlations and by D’haeseleer and Durante (Electron J Combin 27(2):2–32, 2020) for degenerate correlations. In this paper, we completely determine the sets of the absolute points of degenerate correlations, different from degenerate polarities, of a projective space \({{\mathrm{PG}}}(3,q^n)\). As an application we show that, for q even, some of these sets are related to the Segre’s \((2^h+1)\)-arc of \({{\mathrm{PG}}}(3,2^n)\) and to the Lüneburg spread of \({{\mathrm{PG}}}(3,2^{2h+1})\).



中文翻译:

$$ PG(3,q ^ n)$$ PG(3,qn)的绝对相关点

投射空间的(可能是简并的)极性的绝对点集是众所周知的。BC Kestenband在11篇论文中完全确定了((可能退化的)相关性的绝对点集,不同于极性的\({{\ mathrm {PG}}}(2,q ^ n)\)。 1990年至2014年,针对非简并相关,D'haeseleer和Durante(Electron J Combin 27(2):2-32,2020年)针对简并相关。在本文中,我们完全确定了射影空间\({{\ mathrm {PG}}}(3,q ^ n)\)的简并相关绝对点集,与简并极性不同。作为应用程序,我们表明,即使对于q,这些集合中的一些也与Segre的\((2 ^ h + 1)\)- arc有关\({{\ mathrm {PG}}}(3,2 ^ n)\)到Lüneburg的\({{\ mathrm {PG}}}(3,2 ^ {2h + 1})\)

更新日期:2020-08-29
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