Foliated and Mather-Jacobian discrepancies via tangential arcs

Abstract: This article develops a tangential arc-space approach to foliated discrepancies for logarithmic simple co-rank one foliations on threefolds, relative to a fixed invariant normal crossing separatrix divisor. In the non-resonant logarithmic case, reduced tangential arcs centred on the prescribed tangential locus are confined to this divisor. The tangential sector is therefore presented, at the reduced arc level, by the normalised separatrix-conductor system. Foliated adjunction transfers the discrepancy calculus to ordinary log pairs obtained by adjunction on the normalised branches and conductors. The arc-space theorem of Ein-Mustaţă-Yasuda, applied on these strata, then gives a tangential codimension formula identifying logarithmic codimensions of toroidal tangential divisorial cylinders with the corresponding tangential discrepancies. For toroidal invariant divisors read on branches, this tangential discrepancy agrees with the usual foliated discrepancy. The resulting theory gives toroidal tangential inversion of adjunction, a branch--conductor description of the tangential non-lc and non-klt loci, a cylinder criterion for tangential log canonicity, lower semicontinuity of the toroidal tangential minimal log discrepancy, and a relative Mather-Jacobian refinement for the canonical image separatrix system.

• The paper provides an arc-theoretic formulation that transfers discrepancy calculus from foliated threefolds to corresponding Mather–Jacobian log pairs.
• It establishes a branch–conductor presentation for tangential arcs that links arc spaces of normalizations to classical foliated discrepancies.
• The results yield robust criteria for log canonicity, inversion of adjunction, and lower semicontinuity of tangential minimal log discrepancies.

Tangential Arc Spaces and Foliated Discrepancies in Threefolds

Introduction and Motivation

This work presents an in-depth arc-theoretic framework for the analysis of discrepancies in the context of co-rank one foliations on threefolds, extending the classical correspondence between arc space cylinders and discrepancies to the foliated and Mather–Jacobian settings. The central innovation is the construction of a branch–conductor presentation of tangential arc sectors, enabling the transfer of discrepancy calculus on foliations to adjunction pairs on normalizations of invariant divisors and their intersections. The approach builds directly on the arc space techniques of Ein–Mustaţă–Yasuda and the theory of Mather–Jacobian discrepancies as advanced by de Fernex–Docampo, Ishii, and Ein–Ishii.

Tangential Arcs and Non-Resonant Logarithmic Foliations

The setting considered is that of co-rank one logarithmic simple adapted foliations on smooth threefolds, relative to a fixed invariant normal crossing separatrix divisor. Crucially, the non-resonance condition on logarithmic residues ensures that reduced tangential arcs centered at the tangential locus of $\mathcal{G}$ are confined entirely to the separatrix divisor. This arc-confinement property does not hold in general for canonical or log canonical singularities, highlighting the specificity and sharpness of the non-resonant hypothesis.

For such sectors, the reduced tangential arc functor admits a presentation as the ordinary arc spaces of the normalizations of the invariant branches and their conductors. Explicitly, tangential arcs are associated via a coequalizer diagram to arcs on branches and conductor curves, modulo appropriate identifications over conductor strata.

Foliated Adjunction and Discrepancy Transfer

Foliated adjunction is exploited to transfer the discrepancy calculus onto ordinary log pairs arising from the normalizations of the branch–conductor system. The resulting adjunction boundaries on the branches assign coefficient one to all invariant traces and preserve the original coefficients of transverse divisors, together with the conductor different. This boundary is shown to be crepant-compatible under adapted toroidal blowups, ensuring the functoriality of the discrepancy invariants.

A key result is that, for tangential data represented on toroidal invariant divisors with generic branch structure, the tangential discrepancy coincides with the classical foliated discrepancy from the minimal model program (MMP) for foliations.

Ein–Mustaţă–Yasuda Formula and Tangential Minimal Log Discrepancy

A principal theorem is the tangential analog of the Ein–Mustaţă–Yasuda formula: the logarithmic codimension of an adapted toroidal tangential divisorial cylinder equals $q$ times the tangential discrepancy attached to its representing divisor. Notably, the resulting minimal log discrepancy (tmld) in the tangential sector is shown to admit a branch–conductor infimum formula over all relevant adjunction pairs. This, in particular, enables:

• A branch–conductor criterion for tangential log canonicity: the pair is tangentially log canonical along a locus if and only if all branch and conductor adjunction pairs are log canonical along the appropriate fibers.
• Characterization of non-lc and non-klt loci in the tangential sector via the corresponding loci in the adjunction pairs.
• Lower semicontinuity of the adapted toroidal tangential minimal log discrepancy, marking the first arc space lower semicontinuity result for a mld-type invariant in such a foliated sector.

Toroidal Tangential Inversion of Adjunction and Cylinder Criteria

The inversion of adjunction for the tangential sector is established, extending MMP-style foliation adjunction results into the arc-theoretic domain. This enables direct cylinder criteria for tangential log canonicity and log centers via the branch–conductor presentation. The theory partitions the adapted toroidal divisorial cylinders into closed tangential and open transverse sectors, with the former governed by tmld and the latter by the ordinary minimal log discrepancy of the ambient pair.

Relative Mather–Jacobian Refinement

The arc space framework is further refined to accommodate the Mather–Jacobian discrepancy theory. In particular, the construction is made for the canonical image separatrix system on $X$, formed by projecting upstairs branch–conductor systems and taking seminormalization. The Mather–Jacobian discrepancy of a tangential datum is computed intrinsically on this canonical image system, and is independent of the representing model. When the separatrix system is algebraizable, the intrinsic Mather–Jacobian value matches that of the constructed image system.

In the locally complete intersection case with trivial Jacobian correction, the Mather–Jacobian tangential discrepancy coincides with the ordinary tangential discrepancy, providing a precise branch–conductor bridge between these two settings.

Local Model Analysis and Examples

Local computations confirm that invariant branch components do not contribute positively to weighted tangential codimensions due to coefficient-one adjunction cancellation, and only transverse boundary coefficients are relevant. Explicit examples at double and triple intersection points illustrate the decompositions and their ramifications for divisor representation, tangential log canonicity, and the uniqueness of generic branches in irreducible tangential cylinders.

Implications and Further Directions

Formally, this work decouples the computation of tangential discrepancies in the foliated setting from the ambient singularities, reducing it to log pairs on the normalized branches and conductors, where established log discrepancy and Mather–Jacobian machinery is available. This enables transplanting known vanishing theorems (e.g., Du Bois, rationality via MJ criteria) to the tangential sector under suitable hypotheses. Additionally, the functoriality under blowups and model independence within the fixed toroidal category underscores the robustness of the construction.

Potential future developments include the extension to higher co-rank foliations, general resonant cases, and the application of these techniques to study singularities in moduli problems or in the analysis of higher-dimensional foliations. The arc space methodology offers the prospect of unifying various approaches to discrepancy, canonicity, and their singularity-theoretic consequences in algebraic geometry.

Conclusion

This paper provides a comprehensive arc-theoretic treatment of discrepancies for foliated threefolds, embedding the analysis of tangential singularities in the framework of jet schemes and Mather–Jacobian theory. The results furnish canonical, arc-space-formulated criteria for log canonicity, inversion of adjunction, and minimal log discrepancies, specifically suited to the tangential locus of foliated structures with simple non-resonant singularities. The functorial branch–conductor approach enables robust and model-independent calculations, making this framework a significant technical enhancement in the study of birational geometry of foliations and arc spaces on singular spaces (2607.01809).