  A new proof for the embedded resolution of surface singularities in a
three-dimensional smooth ambient space over algebraically closed fields of
arbitrary characteristic. The proof makes use of an upper semicontinuous
resolution invariant which prescribes the center in each step of the resolution
algorithm. The resolution invariant strictly decreases under each blowup. The
definition of the resolution invariant is inspired by the proofs of resolution
of singularities in arbitrary dimension over fields of characteristic zero. It
was developed in an attempt to generalize these proofs to the setting of
arbitrary characteristic. The usual resolution invariant which is used in
characteristic zero behaves very badly over fields of positive characteristic.
It is not upper semicontinuous. Further, the invariant may increase under
blowup which destroys the induction argument. It is shown how these problems
can be overcome in the case of surface singularities which are embedded in a
three-dimensional smooth ambient space. Analogous to the proofs of resolution
of singularities in characteristic zero, coefficient ideals are used to enable
induction on the dimension of the ambient space. Since hypersurfaces of maximal
contact need not exist over fields of positive characteristic, coefficient
ideals are considered instead with respect to all regular hypersurfaces. The
associated invariants are then maximized over all local flags of smooth
subspaces. To construct flags which realize this maximum, techniques are
introduced to maximize invariants of coefficient ideals over all coordinate
changes. These so-called "cleaning techniques" form a core part of the proof.
They can be seen as a characteristic-free generalization of the Tschirnhausen
transformation, a classical technique to construct hypersurfaces of maximal
contact.
