Pre-Tango structures and uniruled varieties
The pre-Tango structure is an ample invertible sheaf of locally exact differentials on a variety of positive characteristic. It is well known that pre-Tango structures on curves often induce pathological uniruled surfaces. We show that almost all pre-Tango structures on varieties induce higher-dimensional pathological uniruled varieties, and that each of these uniruled varieties also has a pre-Tango structure. For this purpose, we first consider the $p$-closed rational vector field induced by a pre-Tango structure, and the smoothness of the fibration induced by the $p$-closed rational vector field.
Moreover, we give two examples: of a $3$-dimensional variety of general type whose automorphism group scheme is not reduced, and of a non-uniruled variety which has a pre-Tango structure inducing a higher-dimensional pathological uniruled variety.