Abstract The article gives a combinatorial characterisation of a ruled cubic surface in PG ( 4 , q ) . In PG ( 4 , q ) , q odd, q ≥ 9 , a set K of q 2 + 2 q + 1 points is a ruled cubic surface if and only if K has class [ q + 1 , 2 q + 1 , 3 q + 1 ] 3 such that every plane that contains four points of K contains at least q + 1 points of K .

中文翻译：

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PG (4, q) 中的类 [q + 1,2q + 1,3q + 1] 3 的集合

摘要 本文给出了 PG (4, q) 中规则三次曲面的组合表征。在 PG (4, q), qodd, q ≥ 9 中，q 2 + 2 q + 1 个点的集合 K 是直纹三次曲面当且仅当 K 具有类 [q + 1, 2 q + 1, 3 q + 1] 3 使得每个包含四个 K 点的平面至少包含 q + 1 个 K 点。