Moduli spaces of sheaves on Fano threefolds and K3 surfaces of genus 9

A complex smooth prime Fano threefold $X$ of genus $9$ is related via projective duality to a quartic plane curve $\mathrm{\Gamma }$. We use this setup to study the restriction of rank $2$ stable sheaves with prescribed Chern classes on $X$ to an anticanonical $K3$ surface $S\subset X$. Varying the threefold $X$ containing $S$ gives a rational Lagrangian fibration $${\mathcal{M}}_S(2,1,7)⇢{\mathbb{P}}^3$$ with generic fibre birational to the moduli space ${\mathcal{M}}_X(2,1,7)$ of sheaves on $X$. Moreover, we prove that this rational fibration extends to an actual fibration on a birational model $\mathcal{M}$ of ${\mathcal{M}}_S(2,1,7)$. In a last part, we use Bridgeland stability conditions to exhibit all $K$-trivial smooth birational models of ${\mathcal{M}}_S(2,1,7)$, which consist in itself and $\mathcal{M}$. We prove that these models are related by a flop, and we describe the positive, movable and nef cones of ${\mathcal{M}}_S(2,1,7)$.