Log Adjunction: effectiveness and positivity

Abstract

This is a first instalment of much larger work about relations between birational geometry and moduli of triples. The extraction of work is mainly related to Theorem 6. It is a weak version of Kawamata's Conjecture 1 and an important technical step toward semiampleness of moduli part of adjunction. To prove Theorem 6, we use relative analogues of b-representations. The proof here is rather complete except for b-mobile property used in Corollary 1.