Sui $\{k , n\}$-archi di un piano grafico finito

Tallini Scafati, Maria

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 40 (1966) no. 3, pp. 373-378

Résumé

In a finite irreducible graphic plane, a $\{k , n\}$-arc is a set $K$ of $k$ points no $n+1$ of which are collinear and containing $n$ collinear points. Denoting by $t_{s}$$s=0,1,\cdots,n$ the number of $s$-secant lines of $K$, we establish a relation connecting the integers $t_{s}$ independent from the two equations already known. Moreover we prove that, if $q > 4$, every $\{k , 3\}$-arc admits some exterior line (i.e. some 0-secant) i.e. $t_{0} > 0$, and we obtain a lower bound for $t_{0}$. If $2 q \le 4$, we determine all the $\{k , 3\}$-arcs free from exterior lines. Finally, we apply these results to the cubic of a Galois plane.