Caps on elliptic quadrics
A \(k\)-cap in \(\mathrm{PG}(n,q)\) is a set of \(k\) points, no three of which are collinear. The maximum size of a \(k\)-cap is only known for a few values of \(n\) and \(q\). In [Can. J. Math. 33, 1299--1305 (1980; Zbl 0449.51002)] \textit{B. E. Kestenband} constructed \((q^{2n + 1} + 1)/(q + 1)\)-caps in \(\mathrm{PG}(2n, q^2)\) which were the intersection of \(2n\) linearly independent Hermitian varieties. In [Can. J. Math. 37, 1163--1175 (1985; Zbl 0571.51002)] the reviewer constructed caps with the same parameters as an orbit under a subgroup of the cyclic Singer group acting on \(\mathrm{PG}(2n, q^2)\). Using the same technique the reviewer also constructed \((q^n + 1)\)-caps in \(\mathrm{PG}(2n - 1,q)\) for any even integer \(n\ge 2\). The equivalence of the \((q^{2n + 1})/ (q + 1)\)-caps in \(\mathrm{PG}(2n, q^2)\) obtained by Kestenband and the reviewer has been established for \(n = 1\) in [\textit{E. Boros} and \textit{T. Szönyi}, Combinatorica 6, 261--268 (1986; Zbl 0605.51008)]. In the paper under review the authors show that the \((q^{2n + 1} + 1)/(q + 1) \)-caps of the reviewer are the intersection of \(2n\) Hermitian varieties for any positive integer \(n \). Coupled with recent work of one of the authors, which show that the caps constructed by Kestenband form an orbit under an appropriate subgroup of a cyclic Singer group, the equivalence of the two constructions for the \((q^{2n + 1} + 1)/(q + 1)\)-caps in \(\mathrm{PG}(2n, q^2)\) has now been established. It is also shown in the paper under review that the reviewer's \((q^n + 1)\)-caps in \(\mathrm{PG}(2n - 1, q)\), for \(n \ge 2\) an even integer, are the intersection of \(n - 1\) linearly independent elliptic quadrics. In fact, for \(n \ge 3\) an odd integer, it is shown that the analogous Singer subgroup orbit of size \(q^n + 1\) is still the intersection of \(n - 1\) linearly independent elliptic quadrics in \(\mathrm{PG}(2n - 1, q)\). However, for odd \(n\), the orbit is a union of lines from a line spread of \(\mathrm{PG}(2n - 1, q)\).
scientific article; zbMATH DE number 828730