Strong resolution of singularities for two-dimensional schemes

Abstract: For varieties over fields of characteristic zero there are very strong results (by Hironaka and several others) on resolution of singularities: a canonical and to a certain extent also functorial procedure by so-called permissible blow-ups.

Overfields of positive characteristic or for schemes with mixed characteristic very little is known, and only in small dimension - more precisely in dimensions 2 and 3. Even in dimension 2, where onehas several methods (e.g., a procedure by Lipman using alternatelynormalization and blow-ups in points), there were until
recently no results on strong resolution as described above.

I will reporton joint work with V. Cossart and S. Saito, in which we fill this gap.We can even treat non-irreducible or non-reducible schemes, as well as 'boundaries' on them.