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

摘要

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 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 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 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)) and hence also to the Tits' ovoid of PG(3, q(n)).