math.AG — Algebraic Geometry
Non-flat toroidal compactifications of the universal abelian variety in genus five and above
We prove that two mixed Delaunay--Voronoi models of the universal abelian variety are not flat in genus five: the classical model over the second-Voronoi compactification and the model over the finite cover that records a symmetric theta divisor. Both counterexamples arise from the same monoid-theoretic obstruction: two inequivalent minimal elements in one fibre-degree slice acquire a common positive translate. The ordinary Delaunay cells and cones are certified by exact all-lattice-point computations; after this geometric input, the obstruction is verified by the integer calculations displayed below. Combining the genus-five example in the fixed-theta model with a theorem of Nakamura--Sugawara gives the sharp threshold: the morphism is flat exactly in genera at most four.