On the multiplicity of terminal singularities on threefolds

TL;DR

This paper calculates the multiplicity of terminal singularities on threefolds, derives optimal inequalities relating multiplicity and index, and applies these results to improve bounds on certain Fano threefolds and extend Fujita freeness conditions.

Contribution

It provides a simple calculation method for terminal singularities' multiplicity and improves existing bounds and conditions in the classification of threefolds.

Findings

Derived explicit multiplicity formulas for terminal singularities

Established optimal inequalities between multiplicity and index

Extended Fujita freeness conditions to nonhypersurface singularities

Abstract

We give the multiplicity of terminal singularities on threefolds by simple calculation. Then we obtain the best inequalities for the multiplicity and the index. By using this, we can improve the boundedness number of terminal weak Q-Fano 3-folds in [KMMT, Theorem 1.2]. Furthermore, we can extended [K, Theorem 3.6] for Fujita freeness conditions to nonhypersurface terminal singularities.