Foliated and Mather-Jacobian discrepancies via tangential arcs

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 shown to be confined to this divisor.

The tangential sector is therefore represented, at the reduced arc level, by the normalised separatrix-conductor system.

Foliated adjunction transfers the discrepancy calculus to ordinary adjunction pairs on the normalised branches and conductors.

Applying the arc-space theorem of Ein-Mustaţă--Yasuda on these strata, this yields a tangential codimension formula identifying logarithmic codimensions of toroidal tangential divisorial cylinders with the corresponding tangential discrepancies.

The resulting theory gives a 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.