Absolute points of correlations of $$PG(4,q^n)$$

Source abstract

Abstract 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 $$\mathrm {PG}(2,q^n)$$ 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. 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 ) have been classified in (Donati and Durante in J Algebr Comb 54:109–133, 2021). In this paper, we consider the four dimensional case and completely determine the sets of the absolute points of degenerate correlations, different from degenerate polarities, of a projective space $$\mathrm {PG}(4,q^n).$$ PG ( 4 , q n ) . As an application, we show that some of these sets are related to the Kantor’s ovoid and to the Tits’ ovoid of $$Q(4,q^n)$$ Q ( 4 , q n ) and hence also to the Tits’ ovoid of $$\mathrm {PG}(3,q^n)$$ PG ( 3 , q n ) .