Normality of general elephants on 3-fold terminal flips

TL;DR

This paper proves the normality of general elephants on 3-fold terminal flips, establishing their Du Val singularities and excluding certain non-Gorenstein singularities on the flipped curve.

Contribution

It demonstrates the normality of general elephants in 3-fold terminal flips and rules out specific non-Gorenstein singularities on the flipped curve.

Findings

General elephants are normal and have Du Val singularities.

No non-Gorenstein singularities of types cE/2, cD/3, cAx/4 on the flipped curve.

Provides foundational results for the singularity structure in 3-fold flips.

Abstract

We prove that the general elephant $E_{X^+}\in |-K_{X^+}|$ is normal where $X--> X^+$ is a 3-fold terminal flip. Hence $E_{X^+}$ has at worst Du Val singularities. As a corollary, there exists no non-Gorenstein singularity of type $cE/2$, $cD/3$, nor $cAx/4$ on the flipped curve $C^+$.