Compactifying the relative Picard functor over degenerations of varieties

TL;DR

This paper introduces a new stability condition for line bundles on degenerating varieties to achieve a compactified relative Picard functor, extending concepts from curves to higher dimensions with asymptotic behavior.

Contribution

It generalizes the balanced multidegree stability from curves to higher-dimensional varieties and proves a unique extension of line bundles in degenerations.

Findings

Defined a new asymptotic stability condition for line bundles.

Proved existence and uniqueness of semistable extensions over degenerations.

Extended the compactification approach from curves to higher-dimensional varieties.

Abstract

Over a family of varieties with singular special fiber, the relative Picard functor (i.e. the moduli space of line bundles) may fail to be compact. We propose a stability condition for line bundles on reducible varieties that is aimed at compactifying it. This stability condition generalizes the notion of `balanced multidegree' used by Caporaso in compactifying the relative Picard functor over families of curves. Unlike the latter, it is defined `asymptotically'; an important theme of this paper is that although line bundles on higher-dimensional varieties are more complicated than those on curves, their behavior in terms of stability asymptotically approaches that of line bundles on curves. Using this definition of stability, we prove that over a one-parameter family of varieties having smooth total space, any line bundle on the generic fiber can be extended to a unique semistable line bundle on the (possibly reducible) special fiber, provided the special fiber is not too complicated in a combinatorial sense.