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

Abstract

A complex smooth prime Fano threefold $X$ of genus $9$ is related via projective duality to a quartic plane curve $Γ$. 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 $$\mathcal{M}_S(2,1,7) \dashrightarrow \mathbb{P}^3$$ 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 $\mathcal{M}_S(2,1,7)$.

Figures (2)

• Figure 1: The $(\beta,\alpha)$-plane in $\mathop{\mathrm{Stab}}\nolimits^+(S)$.
• Figure 2: The positive cone $\overline{\mathop{\mathrm{Pos}}\nolimits}(\mathcal{M}_S)$.

Theorems & Definitions (81)

• Theorem 1.1: = Theorem \ref{['thmresinj']} and Theorem \ref{['thmrestotal']}
• Theorem 1.2: = Corollary \ref{['corratlagfib']}
• Theorem 1.3: = Theorem \ref{['thmactuallagfibrationbiratmodel']}
• Theorem 1.4: = Theorem \ref{['ThmKtrivialBirModels']}
• Lemma 2.1
• proof
• Remark 2.2
• Theorem 2.3: Hoppe's criterion, bibhoppecriterion
• Remark 2.4
• Lemma 2.5