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

TL;DR

This paper explores dual varieties related to key varieties of certain $Q$-Fano 3-folds, providing new geometric interpretations of Sarkisov links and characterizations of specific canonical curves.

Contribution

It introduces dual varieties associated with key varieties of $Q$-Fano 3-folds and interprets Sarkisov links as linear sections of these duals, advancing the understanding of their geometry.

Findings

Dual varieties are introduced and characterized.

Sarkisov links are interpreted as linear sections of dual varieties.

Characterization of a genus 9 canonical curve with a $g_7^2$.

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 $g_{7}^{2}$.