Resolution of degenerate mirror families via toric morphisms

TL;DR

This paper explores resolving degenerate mirror families of Calabi-Yau threefolds through toric morphisms, extending previous methods to more complex singularities and analyzing the geometric implications within toric varieties.

Contribution

It demonstrates that toric morphisms can resolve complex extremal transitions in mirror families, broadening the applicability of Batyrev-Borisov construction.

Findings

Successful resolution of more complicated singularities

Connection established between resolutions and ambient toric geometry

Extension of previous methods to new extremal transitions

Abstract

This paper continues the study of two examples of extremal transitions between families of Calabi-Yau threefolds. In a previous paper we suggested that the "mirror transition" between mirror families predicted by Morrison could be achieved naturally by combining a toric morphism with the Batyrev-Borisov construction. This was carried out for a particular example of a conifold transition. In this paper we show that similar methods work for another extremal transition involving more complicated singularities. We also study how the resolution is related to geometry of the ambient toric varieties, and discuss the connection with recent work by Doran and Harder.