Harbourne–Hirschowitz conjecture on special linear systems of plane curves

Let $\{p_1,\ldots,p_r\}\subset\mathbb{P}^2$ be general points, let $\nu:\widetilde{\mathbb{P}}^2\to\mathbb{P}^2$ be their blow-up, and let $\mathcal{L}=\mathcal{L}_{2,d}(m_1,\ldots,m_r)(p_1,\ldots,p_r)$ be the corresponding linear system. A $(-1)$-curve is a curve $C\subset\mathbb{P}^2$ whose strict transform $\widetilde C$ is a smooth rational curve with self-intersection $-1$. The system is $(-1)$-special if it is $(-1)$-reducible, meaning that

$$\mathcal{L}=\sum_{i=1}^k N_iC_i+\mathcal{M},$$

where $C=\sum_{i=1}^k C_i$ is a configuration of $(-1)$-curves, $\widetilde{\mathcal{M}}\cdot\widetilde C_i=0$ for all $i$, $\operatorname{virtdim}(\mathcal{M})\geq 0$, and $N_i>1$ for at least one $i$. Harbourne–Hirschowitz conjecture. A linear system of plane curves is special if and only if it is $(-1)$-special. The conjecture proposes that all speciality of plane linear systems is explained by repeated $(-1)$-curve components; its general status is open.