The Laface–Ugaglia conjecture on special classes in blow-ups of projective 3-space

Let $X$ be the blow-up of $\mathbb{P}^3$ at a finite number of points in very general position. Write

$$D:=dH-\sum_i m_iE_i$$

for a class in standard form, and let $Q$ be the strict transform of the quadric through the first nine points. Define

$$q(D):=\chi(D_{|Q}).$$

Laface–Ugaglia conjecture. \begin{enumerate} \item If $q(D)\leq 0$, then $h^0(D)=h^0(D-Q)$. \item If $q(D)>0$, then $D$ is special if and only if $d<m_1+m_2-1$, and in that case

$$h^0(D)=\binom{d+3}{3}-\sum_{i=1}^r\binom{m_i+2}{3}+\sum_{m_i+m_j>d+1}\binom{m_i+m_j-d+1}{3}.$$

\end{enumerate} This conjecture gives a criterion and formula for the special linear systems on the blow-up of $\mathbb{P}^3$ at very general points; the first assertion is supported by the preceding proposition under the S.H.G.H. conjecture for ten points in $\mathbb{P}^2$, while the full three-dimensional statement remains open in the source.