In 1998, Thomas Schick discovered a purely homological obstruction to the existence of positive scalar curvature metrics on oriented closed smooth manifolds in terms of torality properties of their fundamental classes. We will combine this obstruction with the enlargeability obstruction to positive scalar curvature metrics introduced by Gromov and Lawson in the 1980s. As an application we construct new examples of manifolds which do not admit positive scalar curvature metrics, but whose Cartesian products admit such metrics. Several fundamental questions remain open. The most important is whether toral manifolds of dimension at least 4 with finite fundamental groups of odd order admit positive scalar curvature metrics.