Duality related with key varieties of ℚ-Fano threefolds constructed from projective bundles

In our previous paper [31], we show that all primeℚ-Fano 3-folds X with only 1/2(1, 1, 1)-singularities in certain 5 classes can be embedded as linear sections into bigger dimensionalℚ-Fano varieties called key varieties; each key variety is constructed from data of the Sarkisov link starting 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 . $g_{7}^{2}.$