Fiber integration of gerbes and Deligne line bundles

Let $\pi :X\to S$ be a family of smooth projective curves, and let $L$ and $M$ be a pair of line bundles on $X$. We show that Deligne’s line bundle $⟨L,M⟩$ can be obtained from the ${\mathcal{K}}_2$-gerbe $G_{L,M}$ constructed in a previous work by the authors via an integration along the fiber map for gerbes that categorifies the well known one arising from the Leray spectral sequence of $\pi$. Our construction provides a full account of the biadditivity properties of $\langle {L,M}\rangle$. The functorial description of the low degree maps in the Leray spectral sequence for $\pi$ we develop are of independent interest, and along the course we provide an example of their application to the Brauer group.