Higher dimensional foliated Mori theory

We develop some basic results in a higher dimensional foliated Mori theory, and show how these results can be used to prove a structure theorem for the Kleiman-Mori cone of curves in terms of the numerical properties of $K_{\mathcal{F}}$ for rank 2 foliations on threefolds. We also make progress toward realizing a minimal model program for rank 2 foliations on threefolds.Comment: 50 pages, new version taking into account referee suggestions, published version to appear in Compositio Mat

Hypersurfaces quasi-invariant by codimension one foliations

We present a variant of the classical Darboux-Jouanolou Theorem. Our main result provides a characterization of foliations which are pull-backs of foliations on surfaces by rational maps. As an application, we provide a structure theorem for foliations on 3-folds admitting an infinite number of extremal rays

Foliation adjunction

We present an adjunction formula for foliations on varieties and we consider applications of the adjunction formula to the cone theorem for rank one foliations and the study of foliation singularities.Comment: 34 page

The implicit construction of multiplicity lists for classes of trees and verification of some conjectures

For the problem of understanding what multiplicities are possible for eigenvalues among real symmetric matrices with a given graph, constructing matrices with conjectured multiplicities is generally more difficult than finding constraining conditions. Here, the implicit function theorem method for constructing matrices with a given graph and given multiplicity list is refined and extended. In particular, the breadth of known circumstances in which the Jacobian is nonsingular is increased. This allows characterization of all multiplicity lists for binary, diametric, depth one trees. In addition the degree conjecture and a conjecture about the minimum number of multiplicities equal to 1 is proven for diametric trees. Finally, an intriguing conjecture about the eigenvalues of a matrix whose graph is a path and its submatrices is given, along with a discussion of some ides that would support a proof of the degree conjecture and the minimum number of 1\u27s conjecture, in general. (c) 2012 Elsevier Inc. All rights reserved

Higher Dimensional Foliated Mori Theory

We develop some foundational results in a higher dimensional foliated Mori theory, andshow how these results can be used to prove a structure theorem for the Kleiman-Mori coneof curves in terms of the numerical properties of $K_{\cal F}$ for rank 2 foliationson threefolds. We also make progresstoward realizing a minimal model program for rank 2 foliations on threefolds

Higher Dimensional Foliated Mori Theory

We develop some foundational results in a higher dimensional foliated Mori theory, andshow how these results can be used to prove a structure theorem for the Kleiman-Mori coneof curves in terms of the numerical properties of $K_{\cal F}$ for rank 2 foliationson threefolds. We also make progresstoward realizing a minimal model program for rank 2 foliations on threefolds

Effective generation for foliated surfaces:Results and applications

We prove some results on the birational structure and invariants of a foliated surface $(X, \mathcal F)$ in terms of the adjoint divisor $K_{\mathcal F}+\epsilon K_X$, 0 < \epsilon \ll 1. Several applications of these ideas are considered as well