The general problem of describing the composition factors and submodule structure of Weyl modules for algebraic groups in positive characteristic is unsolved. We consider Weyl modules with highest weights of a particular form and explain how the Jantzen Sum Formula and certain good filtrations lead to the submodule structures in these cases. The selection of the class of weights was dictated by open problems on incidence matrices in classical finite geometries, and the application of the Weyl module structure to the solution of these problems will also be discussed. This is joint work with Ogul Arslan.