no. 1
Moduli spaces of sheaves on Fano threefolds and K3 surfaces of genus $9$
[Espaces de modules de faisceaux sur les $3$-variétés de Fano et les surfaces K3 de genre $9$]
Mattei, Dominique1
1Bonn Universität, Endenicher Allee 60, 53113 Bonn (Germany)
Annales de l'Institut Fourier, Tome 76 (2026) no. 1, pp. 249-289

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 $\mathcal{M}_S[2,1,3] \dashrightarrow \mathbb{P}^3$ 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 $\mathcal{M}_S[2,1,3]$.

In a last part, we use Bridgeland stability conditions to exhibit all $K$-trivial smooth birational models of $\mathcal{M}_S[2,1,3]$, 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 $\mathcal{M}_S[2,1,3]$.

Une $3$-variété de Fano primitive complexe lisse $X$ de genre $9$ est reliée par dualité projective à une courbe quartique plane $\Gamma $. On utilise cette construction pour étudier la restriction de faisceaux de rang $2$ stables avec classes de Chern prescrites sur $X$ à une surface K3 anticanonique $S\subset X$. Varier la variété $X$ contenant $S$ donne une fibration lagrangienne rationnelle $\mathcal{M}_S[2,1,3] \dashrightarrow \mathbb{P}^3$ avec fibre générique birationnelle à l’espace de modules $\mathcal{M}_X(2,1,7)$ de faisceaux sur $X$. De plus, on prouve que cette fibration rationnelle s’étend en une vraie fibration sur un modèle birationnel $\mathcal{M}$ de $\mathcal{M}_S[2,1,3]$.

Dans une dernière partie, on utilise les conditions de stabilité de Bridgeland pour exhiber tous les modèles birationnels $K$-triviaux de $\mathcal{M}_S[2,1,3]$, qui consistent en lui-même et $\mathcal{M}$. On prouve que ces modèles sont reliés par un flop, et on décrit les cônes positif, mobile et nef de $\mathcal{M}_S[2,1,3]$.

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DOI : 10.5802/aif.3680
Classification : 14D20, 14J60, 14D06, 14F08
Keywords: Moduli spaces of sheaves, K3 surfaces, Fano threefolds, derived categories, Lagrangian fibration, stability conditions
Mots-clés : Espaces de modules de faisceaux, surfaces K3, 3-variétés de Fano, catégories dérivées, fibrations lagrangiennes, conditions de stabilité

Mattei, Dominique  1

1 Bonn Universität, Endenicher Allee 60, 53113 Bonn (Germany)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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     title = {Moduli spaces of sheaves on {Fano} threefolds and {K3} surfaces of genus $9$},
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Mattei, Dominique. Moduli spaces of sheaves on Fano threefolds and K3 surfaces of genus $9$. Annales de l'Institut Fourier, Tome 76 (2026) no. 1, pp. 249-289. doi: 10.5802/aif.3680

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