On the equivalence between the effective adjunction conjectures of Prokhorov-Shokurov and of Li

TL;DR

This paper proves the equivalence between two major conjectures in algebraic geometry related to effective adjunction, utilizing recent advances in the minimal model program and canonical bundle formulas.

Contribution

It establishes the equivalence of the Prokhorov-Shokurov and Li effective adjunction conjectures, connecting their formulations through a uniform rational polytope approach.

Findings

Proves the equivalence of two effective adjunction conjectures.

Introduces a uniform rational polytope for canonical bundle formulas.

Utilizes recent developments in the minimal model program.

Abstract

Prokhorov and Shokurov introduced the famous effective adjunction conjecture, also known as the effective base-point-freeness conjecture. This conjecture asserts that the moduli component of an lc-trivial fibration is effectively base-point-free. Li proposed a variation of this conjecture, which is known as the $\Gamma$-effective adjunction conjecture, and proved that a weaker version of his conjecture is implied by the original Prokhorov-Shokurov conjecture. In this paper, we establish the equivalence of Prokhorov-Shokurov's and Li's effective adjunction conjectures. The key to our proof is the formulation of a uniform rational polytope for canonical bundle formulas, which relies on recent developments in the minimal model program theory of algebraically integrable foliations by Ambro-Cascini-Shokurov-Spicer and Chen-Han-Liu-Xie.