Abstract: The canonical bundle formula for a fibration f from a pair (X,B) to Z is the expression of K_X+B as the pullback of a sum of Q-divisors on Z. More precisely, K_X+B is the pullback of K_Z+D+M where K_Z is the canonical divisor, D carries some informations about singular fibers and M is called moduli part. Prokhorov and Shokurov conjectured that there exists a birational modification Z’ of Z and an integer m, that depends only on the dimension of the fibers and their Cartier index, such thar mM’ is base point free on Z’, where m’ is the moduli part induced by the base change. In this talk we will show how to reduce the…