Absolute points of correlations of $$PG(3,q^n)$$

Abstract

AbstractThe 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 $${{\mathrm{PG}}}(2,q^n)$$ 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 $${{\mathrm{PG}}}(3,q^n)$$ PG ( 3 , q n ) . As an application we show that, for q even, some of these sets are related to the Segre’s $$(2^h+1)$$ ( 2 h + 1 ) -arc of $${{\mathrm{PG}}}(3,2^n)$$ PG ( 3 , 2 n ) and to the Lüneburg spread of $${{\mathrm{PG}}}(3,2^{2h+1})$$ PG ( 3 , 2 2 h + 1 ) .