On the iteration of the adjunction process for surfaces of negative Kodaira dimension (Q1262917)

Let (X,L) be a pair consisting of a smooth, complex, projective surface and L a very ample line bundle on it. Suppose that the Kodaira dimension \(\kappa\) (X) of X is negative. Then using results previously obtained by A. J. Sommese and A. Van de Ven, the authors show that the sectional genus \(g_ k=g_ k(L_ k)\) of the successive iterated minimal reductions \((X_ k,L_ k)\), reach a maximum value and then monotonically decrease to a final value \(g_ n=g(L_ n)\leq g=g(L)\). This result gives an explicit way to express the invariants of any pair (X,L) with \(\kappa (X)<0\), in terms of the invariants of a minimal model.