For a $\mathbb C P^2$ is known a result: if through the generic points $p_1,p_2,\dots p_n$ with multiplicities $m_1,m_2\dots, m_n$ correspondingly a degree $d$ curve passes then $d^2\geq m_1^2+m_2^2+\dots+m_n^2-min (m_i)$. (see Lemma 1 in "Curves in $\mathbb P^2$ and symplectic packings" Geng Xu) The idea of the proof is the following: move slightly a point $p_i$ and apply Bezout theorem.
Can we do the same in $\mathbb CP^3$ ?
For a generic set of points $p_1,p_2,\dots p_n$ with prescribed multiplicities $m_1,m_2\dots m_n$ we are looking for a minimal degree $d$ hypersurface through them. Suppose it is $d$. Then slightly move $p_i$ in different directions and find two surfaces of degree $d$ through new collections of points.
Then apply Bezout theorem of these three hypersurfaces and get $d^3\geq m_1^3+m_2^3\dots +m^2_n- min(m_i)^2$.
The only problem is that these surfaces may intesect by a curve and we can not apply Bezout Theorem. Can we repair this?