Quasi-homogeneous linear systems on P2 with base points of multiplicity 7, 8, 9, 10

In the paper we prove Harbourne-Hirschowitz conjecture for quasi-homogeneous linear systems on ${\mathbb{P}}^2$ for $m=7$, 8, 9, 10, i.e. systems of curves of given degree passing through points in general position with multiplicities at least $m,\mathrm{.}\mathrm{.}\mathrm{.},m,m_0$, where $m=7$, 8, 9, 10, $m_0$ is arbitrary.