  Let S_r be the blow-up of P^2 in r general points, i.e., a smooth Del Pezzo
surface of degree 9-r. For r <= 7, we determine the quadratic equations
defining its Cox ring explicitly. The ideal of the relations in Cox(S_8) is
calculated up to radical. As conjectured by Batyrev and Popov, all the
generating relations are quadratic.
