The fibrewise trivial locus of a line bundle

TL;DR

This paper computes the virtual fundamental class of the locus in the base scheme where a line bundle becomes trivial on fibers, providing a geometric understanding of triviality conditions in algebraic geometry.

Contribution

It introduces a method to compute the virtual fundamental class of the fibrewise trivial locus of a line bundle over smooth schemes in characteristic zero.

Findings

Explicit computation of the virtual fundamental class for triviality loci.

Application to understanding the geometry of line bundle triviality.

Framework applicable to smooth proper schemes over fields of characteristic zero.

Abstract

Let $\mathcal{L}$ be a line bundle on a smooth and proper scheme $X$ over $S$. We compute, in the case where $S$ is smooth over a field of characteristic $0$, the virtual fundamental class of the closed subset of $S$ consisting of those $s\in S$ such that the restriction of $\mathcal{L}$ to the fibre $X_s$ over $s$ is trivial.