Hirschowitz–Harbourne conjecture for special plane linear systems

Let $L=\mathcal{L}_{d}(m_{1},\dots,m_{r})$ be a system of plane curves of degree at most $d$ with multiplicities $m_{1},\dots,m_{r}$ at $r$ general points. Let

$$\pi:\widetilde{\mathbb{P}}^{2}\to\mathbb{P}^{2}$$

be the blow-up of these points, with exceptional divisors $E_{1},\dots,E_{r}$. A $-1$-curve is an irreducible curve $C\subset\mathbb{P}^{2}$ whose proper transform $\widetilde{C}$ has self-intersection $\widetilde{C}^{2}=-1$. Write

$$\widetilde{L}:=\left|d\pi^{*}(\mathcal{O}_{\mathbb{P}^{2}}(1))-\sum_{j=1}^{r}m_{j}E_{j}\right|.$$

Hirschowitz–Harbourne conjecture. The system $L$ is special if and only if there exists a $-1$-curve $C\subset\mathbb{P}^{2}$ such that

$$\widetilde{L}.\widetilde{C}\leq -2.$$

The conjecture gives a geometric characterization of speciality for plane linear systems with general fat points, relating excess dimension to negative intersections with $-1$-curves. Its status is not resolved by the supplied source context.