On Compact Analytic Surfaces, III

In Section 3.3, for each model fibration in Kodaira's list, one constructs an explicit isomorphism Φ: U|Δ*→(Δ*×C)/(Zτ1+Zτ2), determines the multiplicity N of Φ*(dz∧dw) along D, and obtains g(z)=z^{-N-1}k(z) or g(z)=z^{-N}k(z), k(0)=0, according to whether div(Ω)=-D or div(Ω)=0.

“For each member of this list, we then construct an explicit isomorphism Φ : U | Δ * → (Δ * × C)/(Zτ 1 + Zτ 2 ), meaning that both the map Φ and the functions τ 1 , τ 2 will be explicit.This allows us to determine the multiplicity N of Φ * (dz ∧ dw) along D, so that Φ * (dz ∧ dw) = f N h dx ∧ dy with h(p) = 0 in local coordinates (x, y) on U near each point p ∈ D. Thus, we either have g(z) = z -N -”