Elementary links from prime Fano threefolds along two lines

TL;DR

This paper constructs and analyzes elementary links from prime Fano threefolds of specific genera along two lines, revealing their end structures and establishing converses in certain cases.

Contribution

It introduces new elementary links for prime Fano threefolds of genus 12, 10, and 9, detailing their end structures and equivariance properties.

Findings

Links from genus 12 Fano threefolds end with blowups of Fano threefolds of type 2.21.

Links from genus 10 Fano threefolds end with blowups along genus 2 bi-quintic curves.

Links from genus 9 Fano threefolds end with conic bundles over  imes \u0012 with bidegree (3,3).

Abstract

For prime Fano threefolds $X$ of genus $g=12$, $10$ or $9$, and for totally disjoint pairs of lines $Z_1$, $Z_2$ in $X$, we establish links from the blowups of $X$ along $Z_1$ and $Z_2$. If $g=12$, then the links end with the blowups of Fano threefolds of type 2.21 along bi-cubic curves; if $g=10$, then the links end with the blowups of the projectivization of the tangent bundle of the projective plane along genus $2$ bi-quintic curves with a mild condition; if $g=9$, then the links end with conic bundles over the product of two projective lines with the discriminant loci of bidegree $(3,3)$. When $g=12$ or $g=10$, we also establish the converses of the above links. Moreover, we especially focus on the links when $g=12$ and the links are $\mathbb{G}_m$-equivariant.