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 Si (one point blow up of P-2) or P-1 x P-1 by checking only the configuration of lines. Here, we show that we can determine if the disjoint lines produce a contraction to P-1 x P-1 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.