The Adjunction Conjecture and its applications

TL;DR

This paper explores the adjunction conjecture in algebraic geometry, extending known formulas, simplifying proofs of bounds in Fujita's Conjecture, and linking adjunction to isolated log canonical singularities.

Contribution

It extends adjunction formulas for fiber spaces and embeddings, providing new insights and applications in the study of singularities and bounds in algebraic geometry.

Findings

Extended adjunction formulas for fiber spaces and embeddings.

Simplified proof of Kollár's quadratic bound in Fujita's Conjecture.

Connected adjunction concepts to isolated log canonical singularities.

Abstract

We discuss adjunction formulas for fiber spaces and embeddings, extending the known results along the lines of the Adjunction Conjecture, independently proposed by Y. Kawamata and V.V. Shokurov. As an application, we simplify Koll\'ar's proof for the Anghern and Siu's quadratic bound in the Fujita's Conjecture. We also connect adjunction and its precise inverse to the problem of building isolated log canonical singularities.