MMP for co-rank one foliations on threefolds

We prove existence of flips, special termination, the base point free theorem and, in the case of log general type, the existence of minimal models for F-dlt foliated pairs of co-rank one on a $${\mathbb {Q}}$$ Q -factorial projective threefold. As applications, we show the existence of F-dlt modifications and F-terminalisations for foliated pairs and we show that foliations with canonical or F-dlt singularities admit non-dicritical singularities. Finally, we show abundance in the case of numerically trivial foliated pairs.

Invariants of germs of analytic differential equations in the complex plane

We study the classification of germs of differential equations in the complex plane giving a complete set of analytic invariants determining the analytic type of the underlying foliation. In particular we answer in affirmative a conjecture of S. Voronin, and generalize some previous results about dicritical singularities in a straightforward manner. Such problem has its origins in a conjecture proposed by R. Thom in the mid-1970s.