Arcara–Miles conjecture

The conjecture predicts that destabilizations of a line bundle on a smooth projective surface come from objects associated with curves $C$ satisfying $C^2<0$, with an analogous statement for the shifted line bundle. It was formulated by Arcara and Miles in their 2014 work on Bridgeland stability.

Known results

• No negative-self-intersection curves, any Picard rank (Arcara–Miles, 2014).
• Picard rank $2$ with one irreducible negative curve (Arcara–Miles, 2014), including Hirzebruch surfaces.
• Del Pezzo surfaces of Picard rank $3$ (claimed in a 2025 preprint).

August 2026 claimed resolution

On August 26, 2026, the preprint Negative Effective Divisors and Bridgeland Stability of Line Bundles on Surfaces reported a general proof: instability or strict semistability is detected by a nonzero effective Cartier divisor with negative self-intersection. This would establish the Arcara–Miles conjecture, but the preprint is unrefereed and the claim remains unverified.

Current status (as of August 2026): A general solution is claimed by an unrefereed preprint; earlier special cases are established, but the full claim awaits independent verification.