Let Xr be a smooth Del Pezzo surface obtained from P 2 by blowing up r ≤ 8 points in general position. It is well known that for r ∈ {3, 4, 5, 6, 7, 8} the Picard group Pic(Xr) contains a canonical root system Rr ∈ {A2 × A1, A4, D5, E6, E7, E8}. We prove some general properties of the Cox ring of Xr (r ≥ 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 Rr.of the product of two projective spaces |D 1 | × |D 2 | to another projective space |D 1 + D 2 | is well-defined. For this reason, it is much more natural to consider the graded set of projective spaces |Γ(X)| := [D]∈Pic(X) |D| together with all possible morphisms s [D 1 ],[D 2 ] any two effective classes [D 1 ], [D 2 ] ∈ Pic(X).

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