Segre–Harbourne–Gimigliano–Hirschowitz conjecture

Let $f\colon X_r\to \mathbb P^2$ be the blow-up of $\mathbb P^2$ at $r$ very general points, with exceptional divisors $E_1,\ldots,E_r$, and let $H$ be the pullback of the hyperplane class. Write

$$D=dH-\sum_{i=1}^r m_iE_i=(d;m_1,\ldots,m_r),$$

where $m_1\geq m_2\geq\cdots\geq m_r>0$, and define

$$\operatorname{vdim}(D)=\frac{D\cdot(D-K_{X_r})}{2}=\frac{d(d+3)}{2}-\sum_{i=1}^r\frac{m_i(m_i+1)}{2}.$$

SHGH Conjecture. If $r\geq 3$ and $d>m_1+m_2+m_3$, then

$$\dim|D|=\max\{-1,\operatorname{vdim}(D)\}.$$

The conjecture concerns the expected dimension of linear systems on blow-ups of the plane. It has been verified for $r\leq 9$ points, while the general case remains open.