Classifying Q-Fano 3-folds via mirror symmetry

Q-因子末端Fano三重的分类是一个长期悬而未决的问题。20多年前，Reid等人计算了可能的希尔伯特序列，在其间的几年中，大约有100例病例被证明存在。Coates，Corti和Kasprzyk在Fano变种的镜像对称性方面的最新进展提出了一种新的方法：不是直接考虑Fano变种，而是考虑它们可能的环状退化。在这篇论文中，我将从这个角度开始勾画q-因子末端Fano-三重的分类。通过利用新的组合技术，如Laurent倒置，我将对分类做出重大贡献，构建许多明确的例子，并显著提高我们对这些基本变种的理解。

The classification of Q-factorical terminal Fano three-folds is a long-standing open question. The possible Hilbert series were computed over 20 years ago by Reid et al., and in the intervening years approximately 100 cases have been shown to exist. Recent advances in Mirror Symmetry for Fano varieties, developed by Coates, Corti, and Kasprzyk, suggests a new approach: rather than considering the Fano varieties directly, instead consider their possible toric degenerations. In this thesis I will begin to sketch the classification of Q-factorical terminal Fano three-folds from this viewpoint. By exploiting new combinatorial techniques such as Laurent inversion, I will make significant contributions to the classification, constructing many explicit examples and significantly improving our understanding of these essential varieties.