Fano Configurations in Subregular Planes

Abstract

In this article a wide variety of odd-order translation planes, constructedfrom Desarguesian planes by multiple derivation, are shown to admit Fanoconfigurations. For example, we show that subregular planes of odd order $q^{2}$ obtained by the replacement of at most $(q-1)/4$ disjoint nets admitFano configurations. Further, for $s$ odd and greater than 1, any subregularplane of order $q^{2}$ can be embedded into a subregular plane of order $q^{2s}$ that contains the original plane as a subplane and admits Fano configurations.