[Paper Review] Derived Categories of Quadric Fibrations and Intersections of Quadrics

Alexander Kuznetsov|ArXiv.org|Oct 31, 2005

Geometric and Algebraic Topology10 references4 citations

TL;DR

This paper establishes a semiorthogonal decomposition of the derived category of a quadric fibration using the sheaf of even parts of Clifford algebras on the base, generalizing Kapranov's description of a single quadric. It proves that the noncommutative variety $(\mathbb{P}(S^2W^*), B_0)$ is Homologically Projectively Dual to $\mathbb{P}(W)$ under the double Veronese embedding, leading to a full description of the derived category of complete intersections of quadrics via HP-duality.

ABSTRACT

We construct a semiorthogonal decomposition of the derived category of coherent sheaves on a quadric fibration consisting of several copies of the derived category of the base of the fibration and the derived category of coherent sheaves of modules over the sheaf of even parts of the Clifford algebras on the base corresponding to this quadric fibration, generalizing the Kapranov's description of the derived category of a single quadric. As an application we verify that the noncommutative algebraic variety $(\PP(S^2W^*),\CB_0)$, where $\CB_0$ is the universal sheaf of even parts of Clifford algebras, is Homologically Projectively Dual to the projective space $\PP(W)$ in the double Veronese embedding $\PP(W) o \PP(S^2W)$. Using the properties of the Homological Projective Duality we obtain a description of the derived category of coherent sheaves on a complete intersection of any number of quadrics.

Motivation & Objective

• To generalize Kapranov's semiorthogonal decomposition of the derived category of a single quadric to the case of quadric fibrations over a base scheme.
• To construct a semiorthogonal decomposition of the derived category of a quadric fibration in terms of the derived category of the base and the sheaf of even Clifford algebras.
• To establish Homological Projective Duality between $\mathbb{P}(W)$ and the noncommutative variety $(\mathbb{P}(S^2W^*), B_0)$ under the double Veronese embedding.
• To derive a complete description of the derived category of complete intersections of quadrics using the machinery of Homological Projective Duality.

Proposed method

• Use Koszul duality between the coordinate algebra of the quadric fibration and the homogeneous Clifford algebra to construct a semiorthogonal decomposition.
• Define the sheaf of even parts of the Clifford algebra $B_0$ on the base $\mathbb{P}(S^2W^*)$ and show it gives a nontrivial component in the derived category of the fibration.
• Apply the theory of Homological Projective Duality (HP-duality) to the double Veronese embedding $f: \mathbb{P}(W) \to \mathbb{P}(S^2W)$, treating the universal quadric as a hyperplane section.
• Verify that the noncommutative variety $(\mathbb{P}(S^2W^*), B_0)$ satisfies the conditions for HP-duality with $\mathbb{P}(W)$ by checking the required semiorthogonal decomposition of the universal hyperplane section.
• Use the resulting duality to deduce semiorthogonal decompositions for linear sections $X_L$ and $Y_L$, corresponding to complete intersections of quadrics and their noncommutative counterparts.

Experimental results

Research questions

• RQ1 How can the derived category of a quadric fibration over a base scheme be decomposed into simpler triangulated categories?
• RQ2 What is the role of the sheaf of even parts of the Clifford algebra in the derived category of a quadric fibration?
• RQ3 Is the noncommutative variety $(\mathbb{P}(S^2W^*), B_0)$ Homologically Projectively Dual to $\mathbb{P}(W)$ under the double Veronese embedding?
• RQ4 Can the derived category of a complete intersection of quadrics be described via HP-duality and the structure of Clifford algebras?

Key findings

• The derived category of a flat quadric fibration $p: X \to S$ of relative dimension $n-2$ admits a semiorthogonal decomposition: $D^b(X) = \langle D^b(S, B_0), p^*D^b(S)(1), \dots, p^*D^b(S)(n-2) \rangle$, where $B_0$ is the sheaf of even parts of the Clifford algebra on $S$.
• The noncommutative variety $Y = (\mathbb{P}(S^2W^*), B_0)$ is Homologically Projectively Dual to $\mathbb{P}(W)$ under the double Veronese embedding.
• For any subspace $L \subset S^2W^*$ such that $X_L$ is a complete intersection of $r = \dim L$ quadrics in $\mathbb{P}(W)$, there exists a semiorthogonal decomposition $D^b(X_L) = \langle D^b(P(L), B_0), \mathcal{O}_{X_L}(1), \dots, \mathcal{O}_{X_L}(n-2r) \rangle$ if $r \leq n/2$, and dually if $r \geq n/2$
• When $r = \dim L = 1$, $X_1$ is a single quadric and $D^b(P(L), B_0)$ recovers Kapranov’s description: $D^b(X_1) = \langle D^b(B_0), \mathcal{O}_{X_1}(1), \dots, \mathcal{O}_{X_1}(n-2) \rangle$, with $D^b(B_0)$ generated by one or two exceptional objects depending on the parity of $n$.
• When $r = \dim L = 2$, $X_2$ is a smooth complete intersection of two quadrics, and $D^b(X_2)$ admits a semiorthogonal decomposition involving a category $C$ that is either a double cover of $\mathbb{P}^1$ ramified at critical values (even $n$) or a $\mathbb{Z}/2\mathbb{Z}$-stack over $\mathbb{P}^1$ (odd $n$).
• In the case $\dim L = n/2$, there is a fully faithful equivalence $D^b(X_L) \simeq D^b(P(L), B_0)$, showing a duality equivalence between the derived category of the intersection and the noncommutative resolution.