Principalization on logarithmically foliated orbifolds

In characteristic zero, we construct principalization of ideals on smooth orbifolds endowed with a normal crossings divisor and a foliation. We then illustrate how the method can be used in the general study of foliations via two applications. First, we provide a resolution of singularities of Darboux totally integrable foliations in arbitrary dimensions – including rational and meromorphic Darboux foliations. Second, we show how to transform a generically transverse section into a transverse section.