Fiber integration of gerbes and Deligne line bundles

TL;DR

This paper demonstrates how Deligne's line bundle associated with two line bundles on a family of curves can be derived from a $ ext{K}_2$-gerbe using fiber integration, clarifying its biadditivity and functorial properties.

Contribution

It introduces a fiber integration approach for $ ext{K}_2$-gerbes to recover Deligne's line bundle, elucidating biadditivity and functoriality in the context of algebraic geometry.

Findings

Fiber integration of gerbes yields Deligne's line bundle.

The construction clarifies biadditivity properties.

Application to the Brauer group illustrates the functorial maps.

Abstract

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 $\langle{L,M}\rangle$ 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.