Relative canonical sheaves of a family of curves

In this paper we study the relative canonical sheaf of a relatively minimal fibration of curves of genus g⩾2 over a one-dimensional regular scheme. Using the configurations of (−2)-chains, we show that its m-tensored product is base point free for any m⩾2. We utilize Koszul cohomology to prove that the relative canonical algebra of the fibration is generated in degree up to five. It is a generalization of K. Konno's work on the 1-2-3 Conjecture of M. Reid.