Calotte di tipo (m, n) in uno spazio di Galois $S_{r,q}$

Tallini Scafati, Maria

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 53 (1972) no. 1-2, pp. 71-81

Résumé

This paper deals with the study of $\{k,n\}$-caps of a Galois space $S_{r,q}$$(r \ge 3,q > 2)$, that is of the sets of $k$ points of $S_{r,q}$ for which the maximum number of collinear points is $n$. Boundaries for $k$ ensuring that $\{k,n\}$-caps exist are given. Then we get a deeper insight of $\{k,n\}$-caps of the type $(m,n)$, that is, $\{k,n\}$-caps having only $m$-secant and $n$-secant lines. We prove in fact that, if such a cap exists and does not coincide with a hyperplane or with the complementary set of a hyperplane in $S_{r,q}$, then $q$ must be an odd square and $m$, $n$, $k$ are necessarily given by (54), (55), (62) respectively.