The SHGH conjecture for uniform points in the projective plane

Let $R={\mathbb C}[x,y,z]$, let $Y=\{P_1,\ldots,P_r\}$ be a general set of points in $\mathbb{P}^2$, and let $I_Y$ be its defining ideal. An irreducible homogeneous polynomial $F\in R$ is exceptional for $Y$ if, writing $n_i$ for its highest vanishing order at $P_i$, one has

$$\deg(F)^2-\sum_{i=1}^r n_i^2=-3\deg(F)+\sum_{i=1}^r n_i=-1.$$

For $m\in{\mathbb N}$, write $mY$ for the uniform fat point scheme defined by $I_Y^{(m)}$, and let $\operatorname{AH}_2(d)$ denote the expected interpolation condition in degree $d$. SHGH conjecture. The scheme $mY$ is not $\operatorname{AH}_2(d)$ if and only if there exists an irreducible homogeneous polynomial $F\in R$ exceptional for $Y$ such that, for some $s>1$, $F^s$ divides every homogeneous polynomial of degree $d$ in $I_Y^{(m)}$.

This is the uniform-points case of the Segre–Harbourne–Gimigliano–Hirschowitz conjecture, which predicts the exceptional behavior in interpolation by fat points. The source presents it as a proposed route toward determining Hilbert functions of symbolic powers; the general conjecture remains open.