On the Nagata Problem

Nagata has conjectured that the following statement (N_r) holds for all $r\geq 10$: (N_r) if $P_1,...P_r \in {\mathbb P}^2$ are generic points then any plane curve $C$ satisfies $\sum_1^r mult_{P_i}(C)\leq \sqrt{r} deg(C)$. Nagata proved (N_r) whenever $r$ is a perfect square. Here we prove (N_r) provided $r=k^2+\alpha,1\leq\alpha\leq2k,k\geq 3$ and either (i) $\alpha$ is odd and $\alpha\geq \sqrt{2k}$ or (ii) $\alpha$ is even and at lest 6, and the fractional part of $\sqrt{r}$ is at most $2(\sqrt{2}-1)$.