Abstract. We study the projective normality of a minimal surface X which is a ramified double covering over a rational surface S with dim | − K S | ≥ 1. In particular Horikawa surfaces, the minimal surfaces of general type with K 2 X = 2p g (X)−4, are of this type, up to resolution of singularities. Let π be the covering map from X to S . We show that the Z 2 -invariant adjoint divisors K X + rπ * A are normally generated, where the integer r ≥ 3 and A is an ample divisor on S .

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