The Cox ring of a Del Pezzo surface

Let $X_r$ be a smooth Del Pezzo surface obtained from ${\mathrm{\P }}^2$ by blowing up $r \leq 8$pointsingeneralposition\mathrm{.}Itiswellknownthatfor$r \in {3,4,5,6,7,8 }$thePicardgroup$\Pic(X_r)$containsacanonicalrootsystem$R_r \in {A_2 \times A_1, A_4, D_5, E_6, E_7, E_8 }$. We prove some general properties of the Cox ring of $X_r$ ($r\ge 4$) and show its similarity to the homogeneous coordinate ring of the orbit of the highest weight vector in some irreducible representation of the algebraic group $G$ associated with the root system $R_r$.