On the intersection of Hermitian surfaces (Q5902081)

The structure of the intersection of two Hermitian surfaces in \(PG(3,q^2)\) under the hypotheses that in the pencil they generate there is at least one degenerate surface has been determined and it has been proven that under suitable hypotheses the intersection of two Hermitian surfaces generating a non-degenerate pencil is a pseudo-regulus. Let \(H_0\) and \(H_1\) be non-degenerate Hermitian surfaces in \(PG(2,q^3)\) with \(B =H_0 \cap H_1\) the base of the Hermitian pencil they generate. The authors prove that if the Hermitian pencil contains only non-degenerate surfaces then one of the following holds: (1) \(B\) contains exactly two skew lines and \(q^4-1\) other points; (2) \(B\) contains exactly two skew lines, a third line intersecting the two skew lines and \(q^4-q^2\) other points; (3) \(B\) contains exactly four lines forming a quadrangle and \(q^4-2q^2 +1\) other points; (4) \(B\) is ruled by a pseudo-regulus. Additionally they show that each of these cases occur.