Finiteness of numbers of curves on a minimal surface with \(\kappa=1\) (Q1175655)

Let \(S\) be a minimal analytic surface and \(C\) be a curve (i.e. a reduced, irreducible effective divisor on \(S)\). The `arithmetic genus' of \(C\) is the number \(\pi(C)=1+C(C+K_ S)/2\), where \(K_ S\) is the canonical class of \(S\). Fix an integer \(g\) and look to all the algebraic families of curves with arithmetic genus \(g\). Namikawa showed that, modulo the automorphisms of \(S\), there are only finitely many such families, when the Kodaira dimension of \(S\) is different from 1. Using a result of Miyaoka and Umezu, the author shows here that also for surfaces of Kodaira dimension 1, the number of algebraic families of curves with fixed arithmetic genus is finite \((\bmod Aut(S))\).