Scheme of lines on a family of 2-dimensional quadrics: geometry and derived category

TL;DR

This paper constructs a semiorthogonal decomposition of the derived category for a family of 2D quadrics, linking it to resolutions of a double cover and Clifford algebra sheaves, revealing deep geometric and categorical structures.

Contribution

It provides the first semiorthogonal decomposition of the derived category of the relative Fano scheme of lines on a family of quadrics, incorporating resolutions and Clifford algebra sheaves.

Findings

Semiorthogonal decomposition of the derived category of the Fano scheme

Relation between the decomposition and resolutions of singularities

Identification of an orthogonal exceptional collection

Abstract

Given a generic family $Q$ of 2-dimensional quadrics over a smooth 3-dimensional base $Y$ we consider the relative Fano scheme $M$ of lines of it. The scheme $M$ has a structure of a generically conic bundle $M \to X$ over a double covering $X \to Y$ ramified in the degeneration locus of $Q \to Y$. The double covering $X \to Y$ is singular in a finite number of points (corresponding to the points $y \in Y$ such that the quadric $Q_y$ degenerates to a union of two planes), the fibers of $M$ over such points are unions of two planes intersecting in a point. The main result of the paper is a construction of a semiorthogonal decomposition for the derived category of coherent sheaves on $M$. This decomposition has three components, the first is the derived category of a small resolution $X^+$ of singularities of the double covering $X \to Y$, the second is a twisted resolution of singularities of $X$ (given by the sheaf of even parts of Clifford algebras on $Y$), and the third is generated by a completely orthogonal exceptional collection.