The SHGH conjecture for homogeneous linear systems on Hirzebruch surfaces

Let $\mathcal{L}_n(a,b;m_1,\dots,m_r)$ be a linear system on the Hirzebruch surface $\mathbb{F}_n$, with the notation and $-1$-systems defined by the intersection pairing and reduction procedure in the source. A system is $-1$-special when the reduction procedure produces a system $M$ with $\operatorname{edim} M>\operatorname{edim} L$. SHGH conjecture. The system $\mathcal{L}_n(a,b;m_1,\dots,m_r)$ is special if and only if it is $-1$-special. This is the proposed geometric characterization of speciality by fixed $-1$-curves; its status is not resolved in the supplied text.