Consider the Klein quadric \(Q^+(5,q)\) in \(\text{ PG }(5,q)\). 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 \(Q^+(5,q)\). Good quadratic sets can come into 15 possible types and in earlier work we already discussed 11 of these types. The present paper is devoted to the four remaining types. We will describe several infinite families of good quadratic sets of \(Q^+(5,q)\). This will show that there are examples of quadratic sets for each of these four types and for each value of the prime power q.

考虑\(\text{ PG }(5,q)\)中的克莱因二次方程\(Q^+(5,q) \) 。如果\(Q^+(5,q)\)的点集与\(Q^+(5,q)\)的每个平面\(\pi \)以可能可约化的方式相交，则该点集称为二次集。 \(\pi \)的二次曲线，即单例、一条直线、不可约二次曲线、两条直线的铅笔或整个\(\pi \)。如果最多有两种可能性出现在\(\pi \)范围内\(Q^+(5,q)\)的所有平面上，则二次集被称为好的二次集。好的二次集可以有 15 种可能的类型，在早期的工作中我们已经讨论了其中的 11 种类型。本文主要讨论其余四种类型。我们将描述几个良好二次集的无限族\(Q^+(5,q)\)。这将表明这四种类型中的每一种以及素数q的每个值都有二次集的示例。