On cyclic caps in 4-dimensional projective spaces
A cap in a projective space \(\text{PG}(N,q)\) is a set of points no three of which are collinear; a cap is called complete if it is not contained in a larger cap. A cap is said to be cyclic if it is an orbit under a cyclic group of collineations of \(\text{PG}(N,q)\). The complement of a hyperplane in \(\text{PG}(N,q)\) can be naturally identified with the field \(\mathbb F_{q^ N}\) in such a way that the multiplicative group of \(\mathbb F_{q^ N}\) acts as a cyclic collineation group of \(\text{PG}(N,q)\). For \(N=4\), it is known that under this identification the multiplicative group \(C_{q^ 2 +1}\) of order \(q^ 2 +1\) is a cyclic cap. The author shows that this cap is never complete. If \(q\) is odd, \(m \geq 2\) and \(C_{m(q^ 2 +1)}\) is a cap, then \(m =2\) and \(q \equiv 3 \pmod 4\). Moreover, the resulting cap \(C_{2(q^ 2 +1)}\) is complete.
scientific article; zbMATH DE number 5536724