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

TL;DR

This paper explores the geometry of moduli spaces of sheaves on Fano threefolds and K3 surfaces of genus 9, revealing a rational Lagrangian fibration, extending it to an actual fibration, and classifying all K-trivial birational models.

Contribution

It establishes a rational Lagrangian fibration on moduli spaces of sheaves, extends it to a genuine fibration, and classifies all K-trivial birational models using Bridgeland stability.

Findings

Construction of a rational Lagrangian fibration on the moduli space.

Extension of the fibration to a birational model.

Classification of all K-trivial smooth birational models and their relations.

Abstract

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