Log resolution conjecture for schemes in the relevant birational class
Establish that every log pair whose underlying scheme is birational to a normal integral scheme quasi-projective over an excellent ring of finite Krull dimension, admitting a dualizing complex and having no residue characteristics equal to 2, 3, or 5, admits a log resolution obtained by a sequence of regular blowups along the non-simple normal crossing locus at each stage.
References
Conjecture 2.6. Let X be a normal integral scheme quasi-projective over an excellent ring, which is of finite Krull dimension, admits a dualizing complex with no residue characteristic 2, 3 or 5. The log resolution conjecture holds for X if every log pair (Y, ∆Y ) whose underlying scheme Y is birational to X admits a log resolution obtained by a sequence of regular blowups along the non-simple normal crossing locus at each stage.
Our geometric approach via Bertini theorems bypasses this machinery and allows us to work with significantly more general models. Assuming only that the model X is projective, we impose no regularity conditions on it and allow XK to be merely geometrically normal—advantages particularly relevant in dimensions greater than three, where resolution of singularities remains open for models in mixed characteristic and varieties in positive characteristic, respectively.