The SHGH-type conjecture for effective divisors on blow-ups of Hirzebruch surfaces

Let $\mathbb F_{e,r}$ be the blow-up of the Hirzebruch surface $\mathbb F_e$ at $r$ very general points, and let $C_e$ be the negative section of $\mathbb F_e$ with strict transform $\widetilde{C}_e$ on $\mathbb F_{e,r}$. A divisor is non-special when its first cohomology vanishes, and a $(-1)$-curve is a smooth rational curve of self-intersection $-1$. The proposed conjecture. The following holds on $\mathbb F_{e,r}$: (a) effective nef divisors on $\mathbb F_{e,r}$ are non-special; (b) every integral curve $C$ on $\mathbb F_{e,r}$ with $C^2<0$ is either a $(-1)$-curve or the strict transform $\widetilde{C}_e$ of $C_e$. This is an analogue of the SHGH conjecture for blow-ups of $\mathbb P^2$. Part (b) implies bounded negativity, while the conjecture is proved in the paper when $r\leq e+4$; the general case remains open.