GAAG seminar - On the singularities of 1-foliations

In positive characteristic geometry, purely inseparable morphisms can be described in terms of sheaves of tangent vectors called 1-foliations. They have been used by several authors to construct pathological examples, but so far only a few results pertaining to their singularities are available. In this talk, I will present two new results about the local theory of 1-foliations. The first one is the existence of resolutions, up to dimension three and in every positive characteristic. The second one is that the usual MMP notions of singularities, when adapted to 1-foliations, define a surprisingly restricted class of singularities.