We introduce the space $\Pi(G)$ of equivalence classes of $\pi$ -points of a finite group scheme $G$ and associate a subspace $\Pi(G)_M$ to any $G$ -module $M$ . Our results extend to arbitrary finite group schemes $G$ over arbitrary fields $k$ of positive characteristic and to arbitrarily large $G$ -modules, the basic results about “cohomological support varieties” and their interpretation in terms of representation theory. In particular, we prove that the projectivity of any (possibly infinite-dimensional) $G$ -module can be detected by its restriction along $\pi$ -points of $G$ . Unlike the cohomological support variety of a $G$ -module $M$ , the invariant $M \mapsto \Pi(G)_M$ satisfies good properties for all modules, thereby enabling us to determine the thick, tensor-ideal subcategories of the stable module category of finite-dimensional $G$ -modules. Finally, using the stable module category of $G$ , we provide $\Pi(G)$ with the structure of a ringed space which we show to be isomorphic to the scheme $\rm{Proj} \rm{H}^{\bullet}(G,k)$