Elementary links from prime Fano threefolds along two lines
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.