On an efficient induction step with Nklt(X,D)

TL;DR

This paper improves the understanding of the induction step in the study of the 5-canonical map's birationality for projective 3-folds, extending results to lower volume thresholds and providing a 4-dimensional analog.

Contribution

It introduces an alternative approach to the induction step on Nklt(X,D), reducing the volume requirement for birationality results and extending the method to higher dimensions.

Findings

Birationality of the 5-canonical map for volumes > 12^3.

Extension of the method to 4-dimensional case.

Improved volume bounds for effective induction.

Abstract

Applying the effective induction on Nklt(X,D) developed by Hacon, Mckernan and Takayama, Todorov proved that the 5-canonical map is birational for projective 3-folds V with Vol(V) sufficiently large, which was recently improved by Di Biagio in loosing the volume constraint. The observation is that the least efficient induction step can be studied in an alternative way, which allows to assert Todorov's statement for $vol(V)>12^3$. The 4-dimensional analog is also given in this note. The idea works well in all dimensions.