On the projective normality of double coverings over a rational surface

We study the projective normality of a minimal surface X which is a ramified double covering over a rational surface S   with dim⁡|−KS|≥1dim⁡|−KS|≥1. In particular Horikawa surfaces, the minimal surfaces of general type with KX2=2pg(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 Z2Z2-invariant adjoint divisors KX+rπ⁎AKX+rπ⁎A are normally generated, where the integer r≥3r≥3 and A is an ample divisor on S.