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Reducible Curves on Rational Surfaces

Let k be an algebraically closed field of characteristic zero. Throughout the present article we fix k as the ground field. Let X be a smooth projective surface and B a reduced curve on X. Then we define the m-genus Pm[B] and the Kodaira dimension κ[B] as follows (cf. [4]): Let f : V → X be a birational morphism such that the strict transform D = f ′(B) becomes a disjoint union of smooth curves. Then Pm[B] := h0(V ,m(D + KV )) and κ[B] := κ(D + KV ,V ), where KV is the canonical divisor on V and κ(D + KV ,V ) is the (D + KV )-dimension of V (cf. [1]). Note that Pm[B] = P̄m(V − D) and κ[B] = κ̄(V − D), where P̄m(V − D) (resp. κ̄(V − D)) denotes the logarithmic m-genus (resp. the logarithmic Kodaira dimension) of V − D (cf. [1] and [11]). Pairs (X,B) of smooth projective rational surfaces X and irreducible curves B on X were studied from the viewpoint of birational geometry by Iitaka [2, 5], Matsuda [9] and the others. In [3] and [4], Iitaka studied reducible curves B on smooth projective rational surfaces such that #(B) = 2, where #(B) is the number of irreducible components of B. In particular, he proved the following result.