Structure of Rational Open Surfaces with Non{positive Euler Characteristic

We study mainly connected conngurations of irreducible curves r i=1 C i on a nonsingular rational projective complex surface X such that the Euler characteristic (X n r i=1 C i) 0, hereby continuing a former conjecture of the author and work of Gurjar and Parameswaran. Roughly we show that any such connguration can be extended to another connguration s i=1 C i (r s) with still (X n s i=1 C i) 0, such that after appropriate blowing{ups and blowing{downsìn s i=1 C i ' the surface X becomes a ruled surface on which s i=1 C i transforms into one of three possible easy standard conngurations.