An arXiv preprint proposes an exact way to detect when a line bundle fails Bridgeland stability: find a non-zero effective Cartier divisor, the geometric object used to build the relevant subobject, whose self-intersection number is below zero and whose associated subobject meets the required phase comparison. The theorem is stated uniformly for a connected smooth complex projective surface X, real classes B and ω with ω ample, and a line bundle L. It covers both L and the shifted object L[1], with different tests in the two chambers of the stability condition.
A divisor becomes a diagnostic
In the unshifted chamber, where s > 0, the rule is precise. L fails to be stable exactly when there is a non-zero effective Cartier divisor C with negative self-intersection, satisfying s − ω · C > 0, for which the natural subobject L(−C) has a phase at least that of L. Here, phase is the comparison value used by the stability condition. For the less strict semistability condition, failure is marked by a strict phase inequality instead. The result converts a question about possible destabilizing subobjects into a test involving a divisor, a numerical side condition and a phase comparison.
The same question in a second chamber
When s ≤ 0, the object tested is L[1]. It fails to be stable exactly when a non-zero effective Cartier divisor C has negative self-intersection, meets the condition s + ω · C ≤ 0, and produces the divisor-supported subobject L(C)|C with phase at least that of L[1]. For semistability, the phase comparison again becomes strict when stability fails. If B · ω > 0, derived duality reduces the shifted statements to the unshifted result at parameter −B and transforms the resulting divisor subobject into OX(C)|C.
How the argument narrows the search
The proof combines minimal-rank destabilizers, slope Harder–Narasimhan filtrations, the Bogomolov–Gieseker inequality and an ordered Lorentzian partial-sum estimate. Its central reduction is that every minimal positive-rank weak or strict destabilizing subobject has rank one. That rank reduction lets the argument extract the divisor-supported subobject used by the detection theorem.
Two consequences of the test
The paper also describes what strict semistability looks like. If the relevant shift is semistable but not stable, a negative effective Cartier divisor gives a short exact sequence: a divisor-supported subobject, the middle object and the quotient are all semistable and have the same phase. In other words, the borderline failure is represented by a matched-phase decomposition.
A broader geometric corollary follows from the same detection picture. If X contains no integral curve of negative self-intersection, every line bundle is stable for every divisorial stability condition. The statement applies across the surface class specified by the theorem, rather than to a selected example.
Turning phase into numbers
The phase comparisons can be recast as explicit numerical tests. Writing α = c1(L) − B and s = α · ω, stability in the unshifted chamber is characterized by positivity of the stated Θ− expression over all negative effective divisors satisfying the corresponding side condition on C · ω; in the shifted chamber, the analogous Θ+ expression is used with its shifted side condition. At the boundary s = 0, L[1] is stable, while the semistability criteria replace strict positivity by weak inequalities and retain the same side conditions.
A link to scaled stability
Another corollary links the result to a numerical condition called B-twisted dHYM-semistability. Under the same hypotheses, that condition is equivalent to stability of L⊗k under σkB,kω for every positive integer k, and also to stability for all sufficiently large k. The result therefore gives a scaling equivalence between a numerical semistability condition and Bridgeland stability.
A precise result with a clear boundary
The result is a detection statement, not a classification of all destabilizing subobjects: the divisor subobject need not be the particular destabilizer one started with, and its phase is compared only with the target object. It also does not establish a stronger version in which the detecting curve is required to be integral. The preprint therefore stops at detection and does not supply either a full classification or the stronger integral-curve statement.
Paper data and sources
Original title: Negative Effective Divisors and Bridgeland Stability of Line Bundles on Surfaces
Authors: Anthony Mäkelä
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text