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Singular locus of q-logarithmic foliations

Ariel Molinuevo, Federico Quallbrunn

TL;DR

This paper studies the singular locus of generic codimension $q$-logarithmic foliations $\omega$ on smooth varieties and its relation to unfoldings. It combines unfolding theory with the Kupka/persistent singularities framework to show that $\mathrm{Sing}(\omega)$ splits into a Kupka component of codimension $q+1$ and a residual piece, and proves $\mathpzc{Kup}(\omega)=\mathpzc{Per}(\omega)$ under genericity. It provides an explicit algebraic description of the persistent singularities via $\mathscr{I}(\omega)$ (and its closure $\mathscr{K}(\omega)$) equaling $\sum_{|J|=q}\mathcal I(F_{\widehat J})$ and also $\bigcap_{|K|=q+1}\sum_{i\in K}\mathcal I(f_i)$. In projective space, these give a concrete graded ideal description for the scheme of persistent singularities, enabling computable characterizations of higher-codimension foliations.

Abstract

We determine the structure of the singular locus of generic codimension-$q$ logarithmic foliations and its relation with the unfoldings of said foliations. In the case where the ambient variety is the projective space $\mathbb{P}^n$ we calculate the graded ideal defining the scheme of persistent singularities.

Singular locus of q-logarithmic foliations

TL;DR

This paper studies the singular locus of generic codimension -logarithmic foliations on smooth varieties and its relation to unfoldings. It combines unfolding theory with the Kupka/persistent singularities framework to show that splits into a Kupka component of codimension and a residual piece, and proves under genericity. It provides an explicit algebraic description of the persistent singularities via (and its closure ) equaling and also . In projective space, these give a concrete graded ideal description for the scheme of persistent singularities, enabling computable characterizations of higher-codimension foliations.

Abstract

We determine the structure of the singular locus of generic codimension- logarithmic foliations and its relation with the unfoldings of said foliations. In the case where the ambient variety is the projective space we calculate the graded ideal defining the scheme of persistent singularities.
Paper Structure (4 sections, 16 theorems, 73 equations)

This paper contains 4 sections, 16 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Codimension-$q$ foliations
  3. Differential forms with logarithmics poles
  4. Main results

Key Result

Theorem 1

Let $\omega\in H^0(X,\Omega^q_X\otimes \mathcal{L})$ be a generic, locally decomposable, $q$-logarithmic foliation. Then we can decompose its singular locus $\mathrm{Sing}(\omega)$ as the disjoint union of where $\mathpzc{Kup}(\omega)$ is the Kupka variety of $\varpi$ of codim $q+1$ and $\mathcal{H}$ is a variety of dimension $q-1$ or $\mathcal{H}=\emptyset$. Even more so, we have that $\mathpzc{

Theorems & Definitions (49)

  • Theorem 1
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Remark 2.9
  • ...and 39 more