The equivariant orbifold birational classification of toroidal compactifications of tori and semiabelian schemes reduces to finding minimal compactifications in logarithmic geometry, solved combinatorially for algebraic tori, nodal curve Jacobians, and abelian-generic semiabelian schemes.
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Birational Classification of Orbifold Compactified Jacobians
The equivariant orbifold birational classification of toroidal compactifications of tori and semiabelian schemes reduces to finding minimal compactifications in logarithmic geometry, solved combinatorially for algebraic tori, nodal curve Jacobians, and abelian-generic semiabelian schemes.