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Foliated and Mather-Jacobian discrepancies via tangential arcs

This paper establishes a tangential arc-space framework for foliated discrepancies on threefolds, utilizing reduced tangential arcs and the Ein-Mustață-Yasuda theorem to derive a codimension formula that yields a toroidal tangential inversion of adjunction and characterizes non-klt loci via branch-conductor systems.

Original authors: Maurício Corrêa

Published 2026-07-03

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Maurício Corrêa

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are looking at a complex, crumpled piece of paper that has been folded many times. In mathematics, this paper represents a geometric shape called a "threefold" (a 3D space). On this paper, there are invisible lines drawn by a "foliation"—think of these as the grain in a piece of wood or the flow lines of a river. Sometimes, these lines get messy and tangled at specific points, creating "singularities" or knots.

Mathematicians want to measure how "bad" these knots are. They use a tool called discrepancy to measure the severity of the mess. If the mess is too bad, the shape is considered "singular" in a way that breaks certain mathematical rules.

This paper, by Maurício Corrêa, introduces a new, specialized way to measure these knots, specifically when the mess happens right along the "flow lines" of the foliation. Here is the breakdown of the paper's ideas using everyday analogies:

1. The Problem: Measuring the Mess Along the Flow

Usually, when mathematicians measure the messiness of a shape, they look at it from the outside. But sometimes, the mess is hidden inside the flow of the lines.

The Paper's Approach: Instead of looking at the whole 3D shape, the author says, "Let's zoom in on the specific lines where the flow gets stuck." He calls this the tangential arc approach. Imagine tracing a tiny ant walking exactly along the flow lines. If the ant gets stuck or confused, that tells us something about the knot.

2. The Key Trick: The "Invisible Fence" (Non-Resonance)

The paper focuses on a specific type of knot called "logarithmic simple."

The Analogy: Imagine the flow lines are running through a garden with a fence (the invariant divisor). The author proves that if the garden is set up in a specific, non-confusing way (called non-resonant), then any ant trying to walk along the flow must stay inside the fence. It cannot wander off into the open garden.
Why this matters: This is a huge simplification. It means we don't have to check the whole 3D world. We only need to check the fence itself. The complex 3D problem collapses into a simpler 2D problem on the fence.

3. The "Branch and Conductor" Map

Once we know the ants are stuck on the fence, the fence itself might be made of several pieces of wood glued together.

The Branches: These are the individual pieces of wood (the separatrix branches).
The Conductors: These are the glue lines where two pieces of wood meet (the conductor curves).
The Map: The author creates a "map" of this fence system. He shows that to measure the messiness of the whole 3D knot, you just need to measure the messiness on these individual pieces of wood and the glue lines.
The "No Double Counting" Rule: If two pieces of wood meet at a glue line, you don't count the messiness of that glue line twice. The paper provides a strict rule to ensure you count it exactly once, like a fair accounting system.

4. The "Arc Space" Formula

The paper uses a famous mathematical tool (the Ein–Mustaţă–Yasuda formula) which says: "The size of the mess is equal to the length of the path the ant takes before it gets stuck."

The Result: By combining the "fence trick" with the "branch/conductor map," the author creates a new formula. This formula translates the complex 3D messiness directly into a simple calculation on the 2D pieces of wood and glue lines.
The "Log Codimension": This is the paper's way of saying, "How much space does the mess take up?" The paper proves that for these specific types of knots, the answer is always a clean, whole number derived from the 2D map.

5. The "Mather–Jacobian" Refinement (The "Roughness" Check)

Sometimes, the pieces of wood on the fence aren't perfectly smooth; they might be jagged or cracked.

The Analogy: The standard measurement (ordinary discrepancy) assumes the wood is smooth. But if the wood is rough, the ant stumbles more.
The Fix: The author adds a "roughness penalty" called the Mather–Jacobian correction. This measures how jagged the fence itself is. If the fence is smooth, the penalty is zero. If it's jagged, the penalty increases the messiness score.
The Result: The paper shows how to calculate this penalty specifically for the flow lines, ensuring that even if the fence is broken, we get an accurate measurement of the knot.

6. The Big Picture: "Inversion of Adjunction"

In math, "Inversion of Adjunction" is like saying: "If the pieces of the puzzle are perfect, then the whole picture is perfect."

The Paper's Claim: The author proves a "Tangential Toroidal Inversion of Adjunction." In plain English: If you check every single piece of wood and every glue line on the fence, and they all look clean (log canonical), then the entire 3D knot is also clean. You don't need to check the whole 3D shape; checking the 2D map is enough.

Summary of the "Main Results"

Simplification: We can reduce a complex 3D measurement to a 2D measurement on a specific "fence" (invariant divisor).
The Map: We can break this fence down into "branches" (wood) and "conductors" (glue) and measure them individually without double-counting.
The Formula: There is a direct formula linking the "size" of the mess (codimension) to the "severity" of the knot (discrepancy).
The Roughness Check: We can adjust this formula to account for jagged or broken fences using the Mather–Jacobian correction.
The Guarantee: If the 2D map is clean, the 3D knot is clean.

What the paper does NOT do:
The paper is purely theoretical mathematics. It does not claim to solve physical problems in engineering, medicine, or climate science. It does not predict future behaviors of fluids or materials. It is a tool for mathematicians to better understand the geometry of shapes with flow lines, specifically in the realm of "birational geometry" (how shapes can be transformed into one another).

Technical Summary: Foliated and Mather–Jacobian Discrepancies via Tangential Arcs

Problem Statement
This article addresses the computation of foliated discrepancies for co-rank one foliations on threefolds, specifically within the context of the foliated Minimal Model Program (MMP). While the relationship between log discrepancies and the codimensions of divisorial cylinders in arc spaces is well-established for singular varieties (via the theorem of Ein–Mustaţ˘a–Yasuda) and extended to non-locally complete intersection cases via Mather–Jacobian theory (Ishii, Ein–Ishii, de Fernex–Docampo), a corresponding "tangential" analogue for foliations has been lacking.

The core difficulty lies in the nature of foliated singularities. Unlike ordinary singularities, foliated discrepancies depend on the tangency of arcs to the foliation. The paper focuses on logarithmic simple co-rank one foliations on threefolds relative to a fixed invariant normal-crossing separatrix divisor. The challenge is to establish a formula that relates the codimension of tangential arc cylinders to foliated discrepancies, particularly in the presence of non-resonant logarithmic singularities where arcs are confined to invariant divisors.

Methodology
The paper develops a "tangential arc-space approach" that bridges birational geometry and formal curve families. The methodology proceeds through several key stages:

Logarithmic Simple Adapted Setting: The analysis is conducted within a specific class of models derived from Cano's reduction theorem. These are "logarithmic simple adapted" models where the foliation is locally generated by a logarithmic 1-form with positive non-resonant residues. This setting ensures that the foliation admits a resolution to simple singularities via blow-ups centered in the singular locus, providing stable local formal models for computation.
Arc Confinement and Representability: A central technical result (Theorem 4.1) demonstrates that in the non-resonant logarithmic case, reduced tangential formal arcs centered on the prescribed tangential locus are strictly confined to the invariant normal-crossing divisor $D_{inv}$ . Consequently, the reduced tangential arc functor is isomorphic to the reduced arc functor of the invariant divisor. This reduces the problem of computing tangential discrepancies to computing ordinary discrepancies on the invariant divisor.
Branch–Conductor System: To handle the geometry of the invariant divisor (which may have normal crossings), the paper constructs a "normalised separatrix–conductor system." This involves:
- Normalising the irreducible invariant branches ( $S^\nu_\alpha$ ).
- Normalising the pairwise intersection curves, termed conductors ( $C^\nu_{\alpha\beta}$ ).
- Defining a seminormal separatrix space $S^{sn}_W$ via an equaliser of the structure sheaves of the branches and conductors.
  This system allows the tangential sector to be represented by ordinary arc spaces on the normalised branches and conductors.
Foliated Adjunction: The paper utilizes foliated adjunction (Cascini–Spicer, Spicer–Svaldi) to transfer the discrepancy calculus from the threefold to the normalised branches and conductors. This establishes "branch boundaries" ( $\Theta$ ) on the normalised surfaces and "conductor boundaries" on the curves, ensuring crepant compatibility under adapted blow-ups.
Mather–Jacobian Refinement: For singular separatrix strata on the base variety $X$ , the paper introduces a Mather–Jacobian refinement. This involves defining a canonical image separatrix system on $X$ and computing discrepancies relative to this fixed algebraic image, using the formal product of ideals to encode boundaries. This distinguishes the intrinsic formal separatrices from the algebraic image system used for the refinement.

Key Contributions and Results

Tangential Ein–Mustaţ˘a–Yasuda Formula: The paper proves a codimension formula (Theorem 1.4, Corollary 1.4) identifying the logarithmic codimension of adapted toroidal tangential divisorial cylinders with the corresponding tangential discrepancies:
$\text{lcodim}^{\text{tan}}(N^{\text{tan}}_q(E)) = q \cdot a^{\text{tan}}(E; X, \mathcal{F}, \Delta)$
This formula relies on the reduction of the tangential sector to ordinary arc spaces on the branch–conductor system.
Tangential Inversion of Adjunction: A toroidal tangential form of inversion of adjunction is established (Corollary 13.1). It states that a foliated pair is tangentially toroidal log canonical along a closed tangent set $Y$ if and only if every associated adjunction pair on the normalised branch and conductor strata is log canonical along the inverse image of $Y$ .
Characterization of Singular Loci: The paper provides a branch–conductor description of the tangential non-log-canonical (non-lc) and non-kawamata-log-terminal (non-klt) loci (Corollary 13.3). These loci are detected as the union of the images of the corresponding singular loci on the normalised branches and conductors.
Lower Semicontinuity: The adapted toroidal tangential minimal log discrepancy ( $tmld_{tor}$ ) is proven to be lower semicontinuous on the adapted tangential centre (Corollary 13.5). This is derived from the lower semicontinuity of ordinary minimal log discrepancies on the normalised branches (which are surfaces or curves) and the finiteness of the branch–conductor system.
Mather–Jacobian Refinement: The paper defines a tangential Mather–Jacobian discrepancy ( $a^{\text{tan}}_{MJ}$ ) relative to the canonical image separatrix system on $X$ (Definition 15.4, Theorem 15.7). It proves that this refinement detects singularity defects of the separatrix strata that are invisible to the ordinary toroidal tangential discrepancy, particularly when the strata are not local complete intersections.
Cylinder Criterion: A criterion for tangential log canonicity is given in terms of the non-negativity of logarithmic codimensions of tangential cylinders (Corollary 13.6).

Significance and Claims
The paper claims to provide a rigorous framework for computing foliated discrepancies using arc spaces, specifically tailored to the "tangential" nature of foliated singularities. Its significance lies in:

Unification of Perspectives: It combines the infinitesimal viewpoint of Carter's jet schemes (strong tangency, total separatrices) with the birational viewpoint of the foliated MMP (adjunction, inversion of adjunction).
Reduction to Ordinary Theory: By proving that reduced tangential arcs are confined to the invariant divisor in the non-resonant case, the paper reduces complex foliated discrepancy calculations to ordinary arc-space computations on normalised branches and conductors.
Refinement for Singularities: The Mather–Jacobian refinement offers a more precise tool for analyzing singularities where the separatrix strata are not smooth or local complete intersections, extending the applicability of discrepancy theory beyond the locally complete intersection range.
Model Independence: The results are shown to be independent of the choice of adapted toroidal model within the fixed category, relying on the functoriality of the branch–conductor system under adapted blow-ups.

The authors explicitly note that the non-resonance condition is essential for the arc-confinement argument and that the results apply to the "logarithmic simple adapted" sector, which serves as a simplicity assumption in the foliated MMP rather than a mere log canonical assumption. The theory does not claim to cover all foliated singularities (e.g., resonant cases or those requiring non-toroidal resolutions) but provides a complete and effective calculus for the specified class of models.