Logarithmic forms and singular projective foliations

TL;DR

This paper investigates polynomial logarithmic forms defining singular projective foliations, proving their stability in certain cases, identifying new moduli space components, and establishing a foundation for broader stability results.

Contribution

It provides an algebraic proof of infinitesimal stability for codimension two logarithmic foliations and identifies new irreducible components of their moduli space.

Findings

Proved stability of certain logarithmic foliations when q=2.

Discovered new irreducible components of the moduli space.

Showed these components are generically reduced.

Abstract

In this article we study polynomial logarithmic $q$-forms on a projective space and characterize those that define singular foliations of codimension $q$. Our main result is the algebraic proof of their infinitesimal stability when $q=2$ with some extra degree assumptions. We determine new irreducible components of the moduli space of codimension two singular projective foliations of any degree, and we show that they are generically reduced in their natural scheme structure. Our method is based on an explicit description of the Zariski tangent space of the corresponding moduli space at a given generic logarithmic form. Furthermore, we lay the groundwork for an extension of our stability results to the general case $q\ge2$.