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 . $g_{7}^{2}.$

中文翻译：

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对偶性与由射影丛构建的 ℚ-Fano 三重体的关键变体相关

在我们之前的论文 [31] 中，我们证明了所有素数ℚ-Fano 3 倍X在某些 5 类中只有 1/2(1, 1, 1)-奇点可以作为线性部分嵌入到更大维度的ℚ-Fano 簇中，称为关键簇；每个关键变量都是根据 Sarkisov 链路的数据构建的，从 1/2(1, 1, 1) 的一个 1/2(1, 1, 1) 奇点的爆炸开始X。本文介绍了与重点品种相关的品种，这些品种在一定意义上具有双重性。作为一个应用程序，我们解释每个萨尔基索夫链接的基本部分X作为对偶品种的线性部分。在自然环境中描述与关键品种对偶的品种X属 5 具有一个 1/2(1, 1, 1)-奇点，我们还用 a 来表征属 9 的一般正则曲线 G 7 2 。 $g_{7}^{2}.$