Duality Related with Key Varieties of $\mathbb{Q}$-Fano 3-folds. I

Abstract: Abstract. In our previous paper arXiv:2210.16008, we show that any prime $\mathbb{Q}$-Fano 3-folds $X$ with only $1/2(1,1,1)$-singularities in certain 5 classes can be embedded as linear sections into bigger dimensional $\mathbb{Q}$-Fano varieties called key varieties, where each of the key varieties is constructed from certain data of the Sarkisov link staring from the blow-up at one $1/2(1,1,1)$-singularity of $X$. In this paper, we introduce varieties associated with the key varieties which are dual in a certain sense. As an application, we interpret a fundamental part of the Sarkisov link for each $X$ as a linear section of the dual variety. In a natural context describing the variety dual to the key variety of $X$ of genus 5 with one $1/2(1,1,1)$-singularity, we also characterize a general canonical curve of genus 9 with a $g_{7}^{2}$.