The well-known Nagata conjecture states that if \(P_1,\dots,P_r\) are generic points in \({\mathbb P}^2\), then there are no curves of degree \(\delta\) having multiplicity at least \(m\) at each \(P_i\) for \(\delta \leq \sqrt r m\). In other words, if \(I_i\subset k[x_0,x_1,x_2]\) is the homogeneous ideal of \(P_i\), then \(l(2,\delta ,m^r):= \dim (I_1^m\cap\dots\cap I_r^m)_\delta = 0\), for \(\delta \leq \sqrt r m\). Nagata proved this when \(r\) is a square. A similar conjecture in the case when the points are in \({\mathbb P}^d\), \(d\geq 2\), has been stated by Iarrobino:
Let \((r,d)\) be integers with 1) \(d\geq 2\); 2) \(r\geq \max\{d+5,2^d\}\); 3) \((r,d) \notin \{(7,2),(8,2),(9,3)\}\). Then \(l(d,\delta,m^r)=0\), \(\forall \delta < {}^d\sqrt{r}m\).
The author proposes a refinement of this conjecture, by changing the third point in order to get a stronger conclusion:
\(3'\) \((r,d) \notin \{(7,2),(8,2),(9,2),(8,3),(9,3)\}\). Then \(l(d,\delta,m^r)=0\), \(\forall \delta \leq {}^d\sqrt{r}m\).
In the paper the new conjecture is proved for \(r=s^d\), i.e. when the number of the points is a \(d^{th}\) power. As a corollary, new counterexamples for the fourteenth problem of Hilbert can be found (this was actually the original aim for Nagata’s work, and the way this new examples are found follows Nagata’s method). The idea for the proof of the main result relies on deformations, by using collisions of fat points in and special initial ideals in order to be able to convert the statement into a combinatorial one (and then use induction on \(d\)).