The effective-divisor spanning conjecture for moduli spaces on a quadric surface

Let $\xi$ be a Chern character of positive integer rank on $\mathbb{P}^1\times\mathbb{P}^1$, and let $M(\xi)$ be the moduli space of $S$-equivalence classes of semistable sheaves with Chern character $\xi$. A Chern character is above Rudakov's surface when it lies in the region determined by Rudakov's Bogomolov-type inequalities. The construction produces effective Brill–Noether divisors

$$D_V=\{U\in M(\xi):h^1(U\otimes V)\neq 0\}.$$

Effective-divisor spanning conjecture. The method laid out in this paper produces a set of effective divisors spanning the effective cone of $M(\xi)$ for every $\xi$ above Rudakov's surface.

The conjecture is motivated by the fact that the method computes the entire effective cone of the first fifteen Hilbert schemes of points on $\mathbb{P}^1\times\mathbb{P}^1$, the cases to which it was applied. It would provide an algorithmic description of the effective cones of these moduli spaces beyond the computed examples.