The exp-injectivity criterion for fat point schemes in the plane

Let $Z=\sum_i m_iP_i$ be a fat point scheme in $\mathbb{P}^2$ for general points $P_i$, and suppose that $h^1(\mathcal I_Z(k))=0$. Let $\mu_k(Z)$ denote the multiplication map in degree $k$, and say that $\mu_k(Z)$ is exp-injective when it has the expected injectivity behavior. For a curve $C\subset \mathbb{P}^2$, write

$$\widetilde C=dL-\sum_i r_iE_i$$

for its strict transform on the blowup $X$ of the points $P_i$, and let $\delta(C_j,Z,k)$ be the quantity defined in the paper for each component $C_j$. Exp-injectivity conjecture. The map $\mu_k(Z)$ fails to be injective if and only if there exists a curve $C\subset \mathbb{P}^2$ such that $\widetilde C=dL-\sum_i r_iE_i$ has $r_i\leq m_i+1$ and $d\leq k+2$, and

$$C=\sum_j n_jC_j,$$

where each $C_j$ is integral, $\widetilde C_j$ is smooth and rational in $X$, and $\widetilde C_{j_1}\cdot\widetilde C_{j_2}=0$ for distinct components. Moreover,

$$\sum_j\delta(C_j,Z,k)>2l(Z)-k(k+2).$$

This conjecture proposes that failure of injectivity is completely detected by configurations of disjoint smooth rational curves satisfying the stated multiplicity, degree, and numerical conditions.