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

Mattei, Dominique

doi:10.5802/aif.3680

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, Dominique ¹

¹ Bonn Universität, Endenicher Allee 60, 53113 Bonn (Germany)

Annales de l'Institut Fourier, Online first, 41 p.

Résumé
Abstract

Une $3$ -variété de Fano primitive complexe lisse $X$ de genre $9$ est reliée par dualité projective à une courbe quartique plane $Γ$ . 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 $ℳ_{S} [2, 1, 3] ⤏ ℙ^{3}$ avec fibre générique birationnelle à l’espace de modules $ℳ_{X} (2, 1, 7)$ de faisceaux sur $X$ . De plus, on prouve que cette fibration rationnelle s’étend en une vraie fibration sur un modèle birationnel $ℳ$ de $ℳ_{S} [2, 1, 3]$ .

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

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 $K 3$ surface $S \subset X$ . Varying the threefold $X$ containing $S$ gives a rational Lagrangian fibration $ℳ_{S} [2, 1, 3] ⤏ ℙ^{3}$ with generic fibre birational to the moduli space $ℳ_{X} (2, 1, 7)$ of sheaves on $X$ . Moreover, we prove that this rational fibration extends to an actual fibration on a birational model $ℳ$ of $ℳ_{S} [2, 1, 3]$ .

In a last part, we use Bridgeland stability conditions to exhibit all $K$ -trivial smooth birational models of $ℳ_{S} [2, 1, 3]$ , which consist in itself and $ℳ$ . We prove that these models are related by a flop, and we describe the positive, movable and nef cones of $ℳ_{S} [2, 1, 3]$ .

Reçu le : 2023-03-03
Révisé le : 2023-10-26
Accepté le : 2023-12-21
Première publication : 2025-02-10

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

Affiliations des auteurs :

Mattei, Dominique ¹

¹ Bonn Universität, Endenicher Allee 60, 53113 Bonn (Germany)

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Mattei, Dominique. Moduli spaces of sheaves on Fano threefolds and K3 surfaces of genus $9$. Annales de l'Institut Fourier, Online first, 41 p.

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