Negative Effective Divisors and Bridgeland Stability of Line Bundles on Surfaces

Abstract

Let $X$ be a connected smooth complex projective surface. We prove an effective-divisor version of the Arcara--Miles conjecture, together with its strict analogue. For every divisorial Bridgeland stability condition, failure of stability, respectively semistability, of a line bundle or its relevant shift is detected by a natural subobject associated with a non-zero effective Cartier divisor $C$ satisfying $C^2<0$. The proof combines minimal-rank destabilizers, slope Harder--Narasimhan filtrations, the Bogomolov--Gieseker inequality, and an ordered Lorentzian partial-sum estimate that forces the minimal rank to be one. Consequently, strict semistability of a line bundle or its relevant shift is detected by a negative effective divisor, and a numerical semistability condition arising from the twisted deformed Hermitian--Yang--Mills equation is equivalent to stability under all integral scalings.