Birational Classification of Orbifold Compactified Jacobians

TL;DR

This paper classifies orbifold compactifications of algebraic groups and Jacobians using logarithmic geometry, providing solutions for tori, Jacobians of nodal curves, and semiabelian schemes.

Contribution

It reduces the classification problem to a combinatorial problem in logarithmic geometry and solves it for specific algebraic groups and Jacobians.

Findings

Classified orbifold compactifications for algebraic tori and Jacobians.

Reduced the semiabelian case to an open conjecture.

Generalized recent results of Schmitt with geometric interpretation.

Abstract

We study the equivariant orbifold birational classification problem for families of toroidal compactifications of a group $G$ over a toroidal base, in the cases where $G$ is an algebraic torus or a semiabelian scheme. The classification is reduced to the problem of finding the minimal orbifold toroidal compactifications of $G$ in the world of logarithmic geometry, which is shown to be a combinatorial problem. We solve the problem for families of algebraic tori, Jacobians of families of nodal curves, and semiabelian schemes with abelian generic fiber. The general semiabelian case is reduced to an open conjecture. These results generalize and geometrically interpret recent results of Schmitt.