We consider some families of smooth Fano hypersurfaces $X_{ n+2}$ in ${ \bf P}^{ n+2} \times { \bf P}^3$ given by a homogeneous polynomial of bidegree $(1,3)$. For these varieties we obtain lower bounds for the number of $F$-rational points of bounded anticanonical height in arbitrary nonempty Zariski open subset $U \subset X_{ n+2}$. These bounds contradict previous expectations about the distribution of $F$-rational points of bounded height on Fano varieties.
Categories: alg-geom math.AG