# Delaunay decompositions in dimension 4

## TL;DR

This paper proves that in dimensions less than 5, all Delaunay decompositions are simplicially generating, leading to the conclusion that certain stable quasi-abelian schemes are reduced and equivalent in these dimensions.

## Contribution

It establishes that Delaunay decompositions are simplicially generating in dimensions under 5, clarifying the structure of stable quasi-abelian schemes.

## Key findings

- Delaunay decompositions are simplicially generating in dimension less than 5
- Projectively stable quasi-abelian schemes are reduced in these dimensions
- Two classes of stable quasi-abelian schemes coincide in dimension less than 5

## Abstract

The Voronoi cone decompositions has been attracting our attention in the compactification problem of the moduli scheme of abelian varieties. The objects to add as the boundary of the moduli scheme are stable quasi-abelian schemes, reduced or nonreduced, which are described in terms of Delaunay decompositions. In this note, we prove that any Delaunay decomposition is simplicially generating if the dimension is less than 5. As a corollary, if the dimension is less than 5, projectively stable quasi-abelian schemes are reduced, and therefore two kinds of stable quasi-abelian schemes, projectively stable quasi-abelian schemes and torically stable quasi-abelian varieties, are the same class of varieties.