Relative compactification of semiabelian Néron models

Abstract

Let 𝘙 be a complete discrete valuation ring, 𝓴(η) its fraction field, 𝘚 = Spec 𝘙, (𝘎[η], 𝓛[η]) a polarized abelian variety over 𝓴([η]) with 𝓛([η]) symmetric ample cubical and 𝒢 the Néron model of 𝘎[η] over 𝘚. Suppose that 𝒢 is semiabelian over 𝘚. Then there exists a unique relative compactification (𝘗, 𝒩) of 𝒢 such that (α) 𝘗 is Cohen-Macaulay with codim[𝘗](𝘗∖𝒢) = 2 and (β) 𝒩 is ample invertible with 𝒩[𝒢] cubical and 𝒩[η] = [[\otimes]𝓷]𝘓[η] for some positive integer 𝓷. We study the totally degenerate case in [MN24], while we study in [N24] the partially degenerate case and then the case where 𝘙 is a Dedekind domain. Most remarkable is that the compactification satisfying the conditions (α) and (β) is uniquely determined by (𝘎[η], 𝐙𝓛[η]).