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Given a generic family $Q$ of 2-dimensional quadrics over a smooth3-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 adouble covering $X \to Y$ ramified in the degeneration locus of $Q \to Y$. Thedouble covering $X \to Y$ is singular in a finite number of points(corresponding to the points $y \in Y$ such that the quadric $Q_y$ degeneratesto a union of two planes), the fibers of $M$ over such points are unions of twoplanes intersecting in a point. The main result of the paper is a constructionof a semiorthogonal decomposition for the derived category of coherent sheaveson $M$. This decomposition has three components, the first is the derivedcategory 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 bythe sheaf of even parts of Clifford algebras on $Y$), and the third isgenerated by a completely orthogonal exceptional collection.

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