Castelnuovo's conjecture for triple-point linear systems on rational threefolds

Let $W$ be a rational smooth irreducible threefold and let $\mathcal{L}$ be an $r$-dimensional linear system on $W$ such that a general $S\in\mathcal{L}$ is a smooth irreducible surface with $p_g(S)=0$ and $p_a(S)=0$. Let $\mathcal{L}_{\bullet}$ be the sublinear system consisting of surfaces in $\mathcal{L}$ having a triple point at a general point $w\in W$.

Castelnuovo's conjecture. The linear system $\mathcal{L}_{\bullet}$ has dimension $r-10$, and one of the following conditions occurs: (A) a general element $S_{\bullet}\in\mathcal{L}_{\bullet}$ is an irreducible surface whose desingularization $\widetilde{S}_{\bullet}$ is irregular, with $q(\widetilde{S}_{\bullet})=1$, $p_g(\widetilde{S}_{\bullet})=0$, and $p_a(\widetilde{S}_{\bullet})=-1$; (B) the surfaces $S_{\bullet}\in\mathcal{L}_{\bullet}$ are reducible, of the form $S_{\bullet}=F_{\bullet}\cup M_{\bullet}$, where $F_{\bullet}$ and $M_{\bullet}$ are rational surfaces passing through $w$, $M_{\bullet}$ varies with $S_{\bullet}$, $F_{\bullet}$ is fixed, and $F_{\bullet}\cap M_{\bullet}$ is a rational curve; or (C) the surfaces $S_{\bullet}\in\mathcal{L}_{\bullet}$ have the same genera as a general $S\in\mathcal{L}$.

This is a conjectural extension of Castelnuovo's rationality criterion to the behavior of linear systems with triple points on rational threefolds. The alternatives describe the possible geometric behavior of the resulting surfaces, including irregular desingularizations, reducible members, or preservation of the genera.