Extremal-pair existence conjecture for moduli spaces on a quadric surface

Let $\xi$ be a Chern character above the Rudakov $\delta$-surface. A controlling pair is a pair of exceptional bundles associated to $\xi$ that controls the construction of a potential extremal effective divisor; an extremal pair is a controlling pair satisfying the additional conditions required for the associated spectral-sequence resolution. A controlling exceptional bundle is a controlling exceptional bundle of an extremal pair when it occurs in such an extremal pair.

Extremal-pair conjecture. Every $\xi$ above the Rudakov $\delta$-surface has an extremal pair, and every controlling exceptional bundle of an extremal pair is contained in two extremal pairs.

Extremal pairs are intended to identify the Chern characters corresponding to extremal effective divisors produced by this approach. Establishing the conjecture for extremal pairs is described in the source as a current area of research.