Special-case strong Bogomolov–Gieseker conjecture for the classes used in curve counting

Let $X$ be the smooth projective threefold under consideration, let $H$ be its ample divisor, and let $\beta$ and $m$ be the numerical data defining $v_n$. Set

$$b_0=-\frac{n}{2}-\frac{\beta.H}{nH^3},\qquad w_f=\frac{n^2}{4}-\frac{\beta.H}{H^3}-\frac{3m}{nH^3}-\left(\frac{\beta.H}{nH^3}\right)^2.$$

Let the strong Bogomolov–Gieseker conjecture mean the inequality for $\nu_{b,w}$-semistable objects with $\operatorname{ch}_2^{bH}(E).H=(w-\frac{b^2}{2})\operatorname{ch}_0(E)H^3$. Special-case strong Bogomolov–Gieseker conjecture. This conjecture holds when (i) $b=b_0$, some $w<w_f$, and $\operatorname{ch}(E)=v_n$; and it holds in case (ii) when $\beta.H>0$, where

$$b=\operatorname{ch}_2(E).H-\frac{1}{2H^3},\qquad w=b^2+\frac{\operatorname{ch}_2(E).H}{H^3},$$

and $E$ is a torsion-free sheaf satisfying

$$\operatorname{ch}_0(E)=1,\qquad \operatorname{ch}_1(E).H^2=0,\qquad -\operatorname{ch}_2(E).H\in[\beta.H,2\beta.H].$$

These are precisely the special cases proposed as sufficient for the paper’s main curve-counting result; their general validity is not established.