Segre–Harbourne–Gimigliano–Hirschowitz conjecture for plane blowups

Let $X$ be the blowup of $\mathbb{P}^2$ at $n$ very general points, and let $D=dH-\sum_i m_iE_i$ be a divisor class with $d\geq 0$. Call $D$ special when both $h^0(\mathcal{O}(D))$ and $h^1(\mathcal{O}(D))$ are nonzero; otherwise call it nonspecial. A $(-1)$-curve is a smooth rational curve $C\subset X$ with $C^2=-1$. SHGH conjecture. The divisor $D$ is special if and only if it contains a multiple $(-1)$-curve in its base locus. The conjecture predicts an algorithm for the dimensions of linear series on very general plane blowups. It is known for $n\leq 9$ and is most interesting for $n\geq 10$; the source also records that it implies Nagata's conjecture.