The Laface–Ugaglia conjecture for Cremona-reduced linear systems in $\mathbb P^3$

Let $\mathcal L=\mathcal L_{3,d}(m_1,\dots,m_s)$ be a linear system in $\mathbb P^3$ with general base points $p_1,\dots,p_s$. It is Cremona reduced when

$$2d\geq m_{i_1}+m_{i_2}+m_{i_3}+m_{i_4}$$

for every $\{i_1,i_2,i_3,i_4\}\subseteq\{1,\dots,s\}$. Let $K_{\mathbb P^3}$ denote the canonical divisor, and let $\mathcal L\cdot L$ and $Q\cdot(\mathcal L-Q)\cdot(\mathcal L-K_{\mathbb P^3})$ denote the stated intersection numbers. Laface–Ugaglia conjecture. If $\mathcal L$ is Cremona reduced, then $\mathcal L$ is special if and only if either there exists a line $L=\langle p_i,p_j\rangle$ with $\mathcal L\cdot L\leq -2$, or there exists a quadric $Q=\mathcal L_{3,2}(1^9)$ such that

$$Q\cdot(\mathcal L-Q)\cdot(\mathcal L-K_{\mathbb P^3})<0.$$

Equivalently, a Cremona-reduced system is either linearly non-special or contains a quadric surface responsible for its speciality. The conjecture is known when there are at most eight points, and when the points have multiplicities at most four or at most five; it remains open in general.