Pre-Tango structures and uniruled varieties

Yoshifumi Takeda¹

¹ Department of Mathematics Nara Women's University Nara 6308506, Japan

Colloquium Mathematicum, Tome 108 (2007) no. 2, pp. 193-216

Résumé

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.

Détail

DOI : 10.4064/cm108-2-4

Keywords: pre tango structure ample invertible sheaf locally exact differentials variety positive characteristic known pre tango structures curves often induce pathological uniruled surfaces almost pre tango structures varieties induce higher dimensional pathological uniruled varieties each these uniruled varieties has pre tango structure purpose first consider p closed rational vector field induced pre tango structure smoothness fibration induced p closed rational vector field moreover examples dimensional variety general type whose automorphism group scheme reduced non uniruled variety which has pre tango structure inducing higher dimensional pathological uniruled variety