[Paper Review] Derived category of V_{12} Fano threefolds

Alexander Kuznetsov|ArXiv.org|Oct 1, 2003

Algebraic Geometry and Number Theory4 references3 citations

TL;DR

This paper establishes a semiorthogonal decomposition of the derived category of a $V_{12}$ Fano threefold as $\mathcal{D}^b(X) = \langle \mathcal{U}_+, \mathcal{O}_X, \mathcal{D}^b(C^\vee) \rangle$, proving that the nontrivial component is equivalent to the derived category of a genus 7 curve $C^\vee$. The key result is that the Fano surface of conics on $X$ is isomorphic to the symmetric square of $C^\vee$, using kernel functors and orthogonal sections in Lagrangian Grassmannians.

ABSTRACT

A V_{12} Fano threefold is a smooth Fano threefold X of index 1 with Pic X = Z and (-K_X)^3=12. We show that the bounded derived category of coherent sheaves on any V_{12} threefold X admits a semiorthogonal decomposition consisting of two exceptional bundles and of the derived category of a curve of genus 7. As an application we show that the Fano surface of X (the surface parameterizing conics on X) is canonically isomorphic to the symmetric square of the associated genus 7 curve.

Motivation & Objective

• To establish a semiorthogonal decomposition of the derived category of a $V_{12}$ Fano threefold.
• To identify the orthogonal component $\mathcal{A}_X$ in the decomposition as equivalent to the derived category of a curve.
• To demonstrate that the Fano surface of conics on $X$ is isomorphic to the symmetric square of the curve $C^\vee$.
• To use kernel functors and orthogonal sections in Lagrangian Grassmannians to construct and verify the derived equivalence.

Proposed method

• Utilizes the embedding of $X$ into the connected component $\mathsf{LGr}_+(V)$ of the Lagrangian Grassmannian in $\mathbb{P}^9$, with $X = \mathsf{LGr}_+(V) \cap \mathbb{P}^8$.
• Constructs an exceptional pair $(\mathcal{U}_+, \mathcal{O}_X)$ in $\mathcal{D}^b(X)$, leading to a semiorthogonal decomposition $\mathcal{D}^b(X) = \langle \mathcal{U}_+, \mathcal{O}_X, \mathcal{A}_X \rangle$.
• Identifies $\mathcal{A}_X$ as the orthogonal subcategory $^\perp\langle \mathcal{U}_+, \mathcal{O}_X \rangle$.
• Uses the universal family of stable rank 2 vector bundles on $X$ with $c_1=1$, $c_2=5$ to define a kernel functor $\Phi_{\mathcal{E}_1}: \mathcal{D}^b(C^\vee) \to \mathcal{D}^b(X)$.
• Shows that $\Phi_{\mathcal{E}_1}$ is fully faithful and its image lies in $\mathcal{A}_X$, and proves essential surjectivity via restriction to hyperplane sections and gluing arguments.
• Uses the isomorphism of moduli spaces on $K3$ surfaces and the equivalence of kernel functors on $S$ to deduce that $\Phi_{\mathcal{E}_1}$ is an equivalence onto $\mathcal{A}_X$.

Experimental results

Research questions

• RQ1 What is the structure of the derived category of a $V_{12}$ Fano threefold?
• RQ2 How does the orthogonal component $\mathcal{A}_X$ relate to algebraic geometry invariants of $X$?
• RQ3 Is the Fano surface of conics on $X$ isomorphic to a symmetric power of a curve?
• RQ4 Can the derived category of $X$ be described via a curve via semiorthogonal decomposition?
• RQ5 What is the role of the orthogonal section $C^\vee$ in the derived category of $X$?

Key findings

• The derived category of a $V_{12}$ Fano threefold admits a semiorthogonal decomposition $\mathcal{D}^b(X) = \langle \mathcal{U}_+, \mathcal{O}_X, \mathcal{D}^b(C^\vee) \rangle$, where $C^\vee$ is a smooth curve of genus 7.
• The orthogonal component $\mathcal{A}_X$ is equivalent to $\mathcal{D}^b(C^\vee)$, establishing a derived equivalence.
• The Fano surface of conics on $X$ is isomorphic to the symmetric square $S^2C^\vee$ of the genus 7 curve $C^\vee$.
• The kernel functor $\Phi_{\mathcal{E}_1}: \mathcal{D}^b(C^\vee) \to \mathcal{D}^b(X)$ is fully faithful and induces an equivalence onto $\mathcal{A}_X$, with the left adjoint $\Phi_{\mathcal{E}_1}^!$ mapping the structure sheaf of a conic to a length 2 subscheme of $C^\vee$.
• The left adjoint $\Phi_{\mathcal{E}_1}^!$ is isomorphic to the structure sheaf of a length 2 subscheme of $C^\vee$, and the induced map $F_X \to S^2C^\vee$ is an isomorphism.
• The curve $C^\vee$ arises as the orthogonal section $\mathsf{LGr}_-(V) \cap \mathbb{P}^6$ in the dual projective space, and is isomorphic to the moduli space of stable rank 2 vector bundles on $X$ with $c_1=1$, $c_2=5$.