Contractions of del Pezzo surfaces to ℙ² OR ℙ¹ x ℙ¹

In this article, we consider r - 1 disjoint lines given in a del Pezzo surface S_r and study how to determine if a contraction given by the lines produces a map to S₁ (one point blow up of ℙ²) or ℙ¹ x ℙ¹ by checking only the configuration of lines. Here, we show that we can determine if the disjoint lines produce a contraction to ℙ¹ x ℙ¹ by combining a quartic rational divisor class to them. We also study the quartic rational divisor classes along the configuration of lines in del Pezzo surfaces.