Distributions with Unstable Tangent Sheaf on $\mathbb{P}^3$

1. Problem Statement

The paper investigates codimension one distributions on the complex projective space $\mathbb{P}^3$. A distribution is defined as a subsheaf $\mathcal{T}_F \subset \mathcal{T}_{\mathbb{P}^3}$ such that the quotient is torsion-free. The singularities of the distribution occur where the quotient fails to be locally free.

While previous literature (specifically [5, 14]) had established a complete classification of the tangent sheaves for distributions of low degree ($d \le 2$), these cases generally resulted in split or $\mu$-semistable sheaves. The central problem addressed in this work is the classification and characterization of distributions where the tangent sheaf $\mathcal{T}_F$ is nonsplit and unstable.

Specifically, the author seeks to:

1. Relate the instability of $\mathcal{T}_F$ to the existence of subfoliations by curves.
2. Define and utilize the "order of nonstability" to bound the instability.
3. Classify the tangent sheaves of distributions that admit a subfoliation by curves of degree 1 (which corresponds to the maximal possible order of nonstability).

2. Methodology

The author employs a combination of algebraic geometry techniques, specifically focusing on the interplay between sheaf stability, Chern classes, and the geometry of singular schemes.

• Stability and Nonstability Invariants: The paper utilizes the concept of $\mu$-stability (slope stability) for rank 2 reflexive sheaves. For unstable sheaves, the author adopts the invariant "order of nonstability" ($r$) introduced in [26]. This order is defined via the normalization of the sheaf and the existence of global sections in twisted duals.
• Subfoliations by Curves: A key methodological tool is the study of subfoliations. If the tangent sheaf $\mathcal{T}_F$ admits a global section (after twisting), it defines a subdistribution of dimension 1 (a foliation by curves). The author defines an invariant $t_F$ as the minimum integer $k$ such that $H^0(\mathcal{T}_F(k-1)) \neq 0$.
• Chern Class Analysis: The paper rigorously calculates the Chern classes ($c_1, c_2, c_3$) of the tangent sheaf and the associated singular schemes ($Sing_1(F)$ and $Sing_0(F)$) using exact sequences involving the Euler sequence and the structure of the singular locus.
• Classification via Degree: The analysis is stratified by the degree $d$ of the distribution and the degree of the subfoliation. The author leverages existing classifications of foliations by curves of degree 1 (from [14]) to constrain the possible structures of $\mathcal{T}_F$.

3. Key Contributions and Results

A. Characterization of Instability

The paper establishes a precise relationship between the degree of the distribution ($d$), the minimal degree of a subfoliation ($t_F$), and the order of nonstability ($r$).

• Lemma 4.1: For a codimension one distribution of degree $d$ with a nonsplit tangent sheaf $\mathcal{T}_F$, $\mathcal{T}_F$ is unstable if and only if $d \ge 3$ and:
  $$1 \le t_F \le \frac{d - 2 + \varepsilon}{2}$$
  where $\varepsilon \in \{0, 1\}$ and $\varepsilon \equiv d \pmod 2$.
  The order of nonstability is given by $r = \frac{d+\varepsilon}{2} - t_F$.
  Significance: This proves that instability is bounded and directly correlated with the existence of low-degree subfoliations. The "more unstable" the sheaf, the lower the degree of the subfoliation.

B. Classification of Maximal Nonstability (Main Theorem)

The paper provides a complete classification for the case where the order of nonstability is maximal (i.e., $t_F = 1$).

Main Theorem 4.2: Let $\mathcal{F}$ be a codimension one distribution of degree $d \ge 3$ on $\mathbb{P}^3$ with a nonsplit, unstable tangent sheaf of maximal order. Then $\mathcal{T}_F$ belongs to one of two families:

1. Family 1 (Extension of a Line):

   • Chern Classes: $(2-d, 1, d)$.
   • Structure: Generated by an extension of a line $L$.
   • Singular Scheme:
     • $Sing_1(\mathcal{F})$ is a curve of degree $d^2 + 1$ with arithmetic genus $p_a = \frac{d(d-1)}{2}$.
     • $Sing_0(\mathcal{F})$ consists of $d$ collinear points.
   • Existence: The author proves that every rank 2 reflexive sheaf generated by such an extension is the tangent sheaf of a valid distribution.

2. Family 2 (Extension of Skew Lines or Double Line):

   • Chern Classes: $(2-d, 2, 2d)$.
   • Structure: Generated by an extension of a pair of skew lines (or a double line of genus -1).
   • Singular Scheme:
     • $Sing_1(\mathcal{F})$ is a complete intersection curve of degree $d^2$.
     • $Sing_0(\mathcal{F})$ consists of two pairs of $d$ collinear points (or $d$ points on the double line).
   • Existence: While not every sheaf in this family is guaranteed to be a tangent sheaf, the author constructs explicit examples of homogeneous 1-forms that realize these singular schemes.

C. Non-Existence of Logarithmic Foliations

In the final section, the author investigates whether these unstable distributions can be logarithmic foliations.

• Lemma 5.1: It is proven that no generic logarithmic foliations exist within these two families of maximal nonstability.
• Reasoning: The degree of the singular curve for generic logarithmic foliations is bounded by $\frac{1}{2}(d+1)^2$, which is strictly less than the required degree ($d^2$ or $d^2+1$) for the distributions in the Main Theorem (for $d \ge 3$).

4. Significance

• Bridging Stability and Geometry: The paper successfully links the abstract algebraic property of sheaf instability to concrete geometric features (the degree and structure of subfoliations and singular schemes).
• Completing the Classification: It fills a gap in the classification of distributions on $\mathbb{P}^3$ by addressing the previously unclassified regime of nonsplit, unstable sheaves ($d \ge 3$).
• Moduli Space Implications: By characterizing the tangent sheaves, the work contributes to understanding the geometry of the moduli space of distributions, specifically identifying components where the tangent sheaf is unstable.
• Explicit Constructions: The paper moves beyond theoretical existence by providing explicit examples (Examples 3, 4, and 5) of distributions with these specific unstable properties, including a locally free but unstable example.

In summary, Barbassa demonstrates that the instability of the tangent sheaf is not a random phenomenon but is tightly constrained by the degree of the distribution and the geometry of its singular locus, culminating in a precise classification of the "most unstable" cases.