D$_4$-flops of the E$_7$-model

TL;DR

This paper investigates the structure of crepant resolutions of E7-models in algebraic geometry, explicitly constructing some resolutions and analyzing their flop diagrams related to Dynkin diagrams.

Contribution

It explicitly constructs four crepant resolutions of E7-models and elucidates their flop relations using simpler models, advancing understanding of their geometric structure.

Findings

Constructed four crepant resolutions of E7-models.

Identified the flop diagram as a D4 sub-diagram.

Connected flops to resolutions of suspended pinch points.

Abstract

We study the geography of crepant resolutions of E$_7$-models. An E$_7$-model is a Weierstrass model corresponding to the output of Step 9 of Tate's algorithm characterizing the Kodaira fiber of type III$^*$ over the generic point of a smooth prime divisor. The dual graph of the Kodaira fiber of type III$^*$ is the affine Dynkin diagram of type E$_7$. A Weierstrass model of type E$_7$ is conjectured to have eight distinct crepant resolutions whose flop diagram is a Dynkin diagram of type E$_8$. We construct explicitly four of these eight crepant resolutions forming a sub-diagram of type D$_4$. We explain how the flops between these four crepant resolutions can be understood using the flops between the crepant resolutions of two well-chosen suspended pinch points.