Boundedness of Fano threefolds with log-terminal singularities of given index

TL;DR

This paper establishes the boundedness of Fano threefolds with log-terminal singularities of fixed index, extending previous results by removing the Q-factoriality and Picard number constraints through advanced minimal model techniques.

Contribution

It proves a boundedness theorem for Fano threefolds with log-terminal singularities of any fixed index, improving earlier results by relaxing key assumptions.

Findings

Boundedness of Fano threefolds with fixed index proven.

New techniques include minimal model program with boundary, Kollár's base point freeness, and Kawamata's extremal curve results.

Extended boundedness results beyond Q-factorial and Picard number 1 cases.

Abstract

We prove the boundedness theorem for Fano threefolds with log-terminal singularities of any fixed index. This is an improvement of our earlier result, where we required additionally that the variety is Q-factorial, with Picard number 1. The new ideas of the paper include the following. 1. Using Alexeev Minimal Model program with suitable boundary to find horizontal extremal contractions. 2. Using Koll\'ar's effective Base Point Freeness theorem. 3. Using Kawamata's result on the length of extremal curves with suitable boundary to avoid gluing curves in some cases.