On the Nef cones of blowups of the projective plane

In this paper, we study the Cremona action on the nef cone of $S_n$, the blowup of ${\mathrm{\P }}^2$ in $n$ very general points, with $n\ge 9$. We construct and describe a rational polyhedral fundamental domain of the nef cone for $n=9$ with respect to this action. In the case $n\ge 10$, we give a rational polyhedral fundamental domain of the $K_{S_n}$-negative part of the nef cone with respect to the Cremona action.