The space of foliations on projective spaces in positive characteristic

Technical Summary: The Space of Foliations on Projective Spaces in Positive Characteristic

Problem Statement
The study of the space of holomorphic foliations on projective spaces has been a central topic in complex algebraic geometry, notably following the foundational work of Jouanolou, Cerveau, and Lins Neto. While the irreducible components of these spaces are well-understood over the complex numbers for low degrees (specifically degrees zero, one, and two), the behavior of these spaces over algebraically closed fields of positive characteristic remains largely unexplored. This paper addresses the classification of the irreducible components of the scheme $\text{Fol}^1_d(\mathbb{P}^n_k)$, representing codimension one foliations of degree $d$ on $\mathbb{P}^n_k$, where $k$ is an algebraically closed field of characteristic $p > 0$. The primary objective is to determine how the geometry and the number of irreducible components of these spaces vary with the characteristic $p$, particularly in low degrees ($d=0, 1, 2$).

Methodology
The authors employ a combination of algebraic geometry techniques specific to positive characteristic and adaptations of complex-geometric arguments. Key methodological elements include:

1. Euler's Relation and Closed Forms: The analysis of degree zero foliations relies heavily on Euler's relation for homogeneous forms. The authors utilize the specific behavior of closed projective forms in characteristic $p$, noting that closed forms of degree $e$ exist only if $p$ divides $e$ (or under specific conditions for exactness).
2. Medeiros' Classification: The paper adapts Medeiros' description of locally decomposable integrable differential forms with linear coefficients. While originally formulated over $\mathbb{C}$, the authors verify its validity in arbitrary characteristic, using it to classify degree zero and one foliations.
3. $p$-Curvature and Degeneracy Divisors: For degree one foliations, the authors utilize the theory of $p$-curvature. They distinguish between $p$-closed and $p$-dense foliations, analyzing the degeneracy divisor $\Delta_F$ associated with the $p$-curvature morphism. This allows them to characterize foliations as linear pull-backs or logarithmic forms based on the degree relative to $p$.
4. Cartier Transform: A central tool for the study of logarithmic components is the Cartier transform of foliations. By applying the Cartier operator to closed logarithmic 1-forms, the authors analyze the stability of these forms under deformation and establish conditions under which logarithmic components remain irreducible.
5. Arithmetic Specialization: To prove results for arbitrary characteristic, the authors often establish results in characteristic zero (or large $p$) and use semi-continuity arguments or specialization to deduce properties in small characteristic. Conversely, they use the existence of specific components in characteristic $p$ to provide alternative proofs for classical results over $\mathbb{C}$.

Key Contributions and Results

• Degree Zero Foliations ($d=0$):
  The authors provide a complete description of the irreducible components for degree zero foliations.

  • If $\text{char}(k) \neq 2$ or the codimension $q > 1$, the space is isomorphic to the Grassmannian of linear subspaces, corresponding to foliations defined by linear projections.
  • In the exceptional case where $\text{char}(k) = 2$ and $q=1$, the space corresponds to the projective space of closed 1-forms of degree 2, which differs significantly from the characteristic $\neq 2$ case.

• Degree One Foliations ($d=1$):
  The number of irreducible components depends strictly on the characteristic $p$:

  • $p=2$: There is exactly one irreducible component, the linear pull-back component $\text{Lin}_1(\mathbb{P}^n_k)$.
  • $p=3$: There are exactly two components: the linear pull-back component $\text{Lin}_1(\mathbb{P}^n_k)$ and the component of closed 1-forms $\text{Cl}_1(\mathbb{P}^n_k)$.
  • $p \geq 5$: There are exactly two components: $\text{Lin}_1(\mathbb{P}^n_k)$ and the logarithmic component $\text{Log}_{(1,2)}(\mathbb{P}^n_k)$.
    The paper establishes that for $p \geq 5$, the description is uniform, and the logarithmic component arises from forms with two distinct residues.

• Degree Two Foliations ($d=2$):
  The results for degree two are less definitive but provide a partial classification.

  • The authors show that the six irreducible components known in characteristic zero specialize to irreducible components in characteristic $p$ for $p \geq 5$.
  • They identify specific components for small characteristics ($p=2, 3$), including the "exceptional" component $\text{Exc}(\mathbb{P}^n_k)$, which is defined by the orbit of a foliation associated with a non-abelian Lie subalgebra of vector fields.
  • The existence of the exceptional component is proven for $p \neq 3$. For $p=3$, the status of this component remains an open question due to the failure of the non-Kupka singularity condition required by their main technical tool.

• Theorem A (Stability of Logarithmic Forms):
  A major theoretical contribution is Theorem A, which establishes the irreducibility of logarithmic components $\text{Log}_{(d_1, \dots, d_r)}(\mathbb{P}^n_k)$ in arbitrary characteristic. The theorem provides specific arithmetic conditions on the degrees $d_i$ and the characteristic $p$ (e.g., whether $p$ divides the degrees or their sums) under which these sets form irreducible components. This result generalizes Calvo-Andrade's stability theorem for generic logarithmic 1-forms from the complex setting to arbitrary characteristic.

Significance and Claims
The paper claims to initiate the systematic study of foliation spaces in positive characteristic, revealing that the geometry of these moduli spaces is highly sensitive to the characteristic of the base field.

• Arithmetic Perspective: The authors demonstrate that Theorem A offers a new, arithmetic proof of Calvo-Andrade's stability result over $\mathbb{C}$ (Corollary 6.3), derived from the behavior of these spaces in positive characteristic.
• Modesty on Completeness: The authors explicitly state that while they provide a complete description for degrees zero and one, the classification for degree two remains incomplete, particularly for small characteristics ($p=2, 3$) and for higher codimensions. They note that the situation for degree three foliations is even less understood.
• Open Questions: The paper concludes by highlighting open problems, such as the completeness of the list of components for degree two in small characteristic and the classification of irreducible components for higher degrees and codimensions.

In summary, the work establishes that while the complex classification serves as a baseline, the positive characteristic setting introduces distinct phenomena (such as the emergence of closed forms in characteristic 2 and the dependence of component counts on $p$) that require specific arithmetic tools for their analysis.