Quadratic sets on the Klein quadric

Consider the Klein quadric Q(+)(5, q) in PG(5, q). A set of points of Q(+)(5, q) is called a quadratic set if it intersects each plane pi of Q(+)(5, q) in a possibly reducible conic of pi, i.e. in a singleton, a line, an irreducible conic, a pencil of two lines or the whole of pi. A quadratic set is called good if at most two of these possibilities occur as pi ranges over all planes of Q(+)(5, q). We obtain several classification results for good quadratic sets. We also provide a complete classification of all good quadratic sets of Q(+)(5, 2) and give an explicit construction for each of them. (c) 2022 Elsevier Inc. All rights reserved.