Letters of a Bi-Rationalist IV: Geometry of log flips

TL;DR

This paper explores the geometric properties of log flips in birational geometry, focusing on inequalities involving the exceptional locus and minimal log discrepancy, and discusses conjectures and examples in the context of log Fano contractions.

Contribution

It investigates a conjectured inequality related to log flips, connecting flip existence, minimal log discrepancy, and dimension, with results in low dimensions and specific cases.

Findings

Conjecture holds in dimensions up to 4

Examples illustrate the conjectured inequality

Connections between flip existence and minimal log discrepancy

Abstract

For a birational log Fano contraction, it is conjectured an inequality between the dimension of its exceptional locus and the minimal log discrepancy over the locus. The conjecture follows from the existence of the flip for the contraction and another conjecture about the maximum of minimal log discripancy in given dimension; in particular, it holds in small dimensions (up to 4), and some special cases. It is illustrated by some examples.