Symbolic powers of ideals of generic points in \(\mathbb P^3\) (Q456827)

Let \(k\) be a field of characteristic zero and \(k[\mathbb{P}^N]=k[x_0,x_1,\ldots,x_N]\) be the ring of coordinates of the projective plane with the standard grading. Let \(I\subseteq k[\mathbb{P}^N]\) be an ideal of fat points in \(\mathbb{P}^N\) and \(M=(x_0,x_1,\ldots,x_N)\subseteq k[\mathbb{P}^N]\) the maximal homogeneous ideal. Harbourne and Huneke conjectured that \(I^{(Nr)}\subseteq M^{(N-1)r}I^r\), for all \(r\geq1\). The conjecture is known to be true for ideals of generic points in \(\mathbb{P}^2\). The author shows that is also true for any number of generic points in \(\mathbb{P}^3\) and for up to \(N+1\) generic points in \(\mathbb{P}^N\), for all \(N\geq2\). As a consequence, he also shows that the conjecture of Chudnovski holds for any number of generic points in \(\mathbb{P}^3\).