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

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${}^{\ast }$ over the generic point of a smooth prime divisor. The dual graph of the Kodaira fiber of type III${}^{\ast }$ 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.

35 pages + one appendix and references

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