This paper expands on a recent remarkable paper in the subject of combinatorial compactifications of the moduli space of abelian varieties [V. Alexeev and A. Brunyate, Invent. Math. 188, No. 1, 175–196 (2012; Zbl 1242.14042)]. Pioneering work of Mumford and others from the 70s showed how to apply the general theory of toroidal embeddings, a method of compactifying a variety closely related to – in fact, étale locally modelled on – the way a toric variety uses a fan to compactify a torus, to the moduli space \(\mathcal{A}_g\) of \(g\)-dimensional principally polarized varieties. The combinatorics in this case concern the space of rank \(g\) real positive definite quadratic forms (more precisely, its “closure” given by positive semi-definite forms with null space defined over \(\mathbb{Q}\)). Since the Deligne-Mumford compactification \(\overline{\mathcal{M}}_g\) of the moduli space of curves is toroidal, the relation between this stack and these toroidal compactifications of \(\mathcal{A}_g\) then becomes a combinatorial game. For instance, the classical Torelli map \(\mathcal{M}_g \rightarrow \mathcal{A}_g\) sending a curve to its (principally polarized) Jacobian can be enlarged to a morphism from the moduli of stable curves to a particular compactification of the moduli of abelian varieties if and only if the combinatorics of the latter are sufficiently compatible with those of the former.
There are three known toroidal compactifications of \(\mathcal{A}_g\) in general, two of which are the key characters in the present paper: the perfect cone compactification (sometimes called “first Voronoi”) and the 2nd Voronoi compactification. Mumford and Namikawa gave a combinatorial argument that for the latter one indeed has an extension of the Torelli morphism. Amazingly, the combinatorics required to address this question in the former case remained unsolved until the very recent above cited work of Alexeev-Brunyate (written at the beginning of Brunyate’s graduate studies under Alexeev) where they showed that indeed there is an extension (and, remarkably, for the third compactification, the “central cones”, there is not for large enough genus!). Additionally, in their paper, they provided a combinatorial criterion for two toroidal compactifications of \(\mathcal{A}_g\) to coincide on a Zariski open subset containing the image of \(\overline{\mathcal{M}}_g\) and showed that this is satisfied in the case of the first and second Voronoi. At the conclusion of their paper, they mention that the combinatorial methods they introduce and use in their proof might extend to a larger, so called “matroidal” locus, thus giving a larger open subset on which the two compactifications agree.
The authors of the present paper interpret this remark as a conjecture and show that it indeed holds. In fact, they show that in a certain sense this matroidal locus is the “largest” locus where the compactifications coincide, suitably interpreted. Due to the machinery developed by Mumford et al., and further developed in this specific setting by Alexeev-Brunyate, there is absolutely no geometry required in the present paper: it truly is a purely combinatorial paper which has a very nice geometric consequence. The key ingredient in their proof is a combinatorial result due to Seymour from 1980 concerning decompositions of regular matroids – a result that makes no mention of abelian varieties whatsoever. Indeed, the authors argument is in essence a reduction to (a slight modification of) Seymour’s result that allows them to bootstrap up from rather elementary/well-known combinatorial statements to the ones needed to establish the Alexeev-Brunyate result.
The paper begins with a very elementary and readable exposition of the combinatorial objects involved in these toroidal compactifications, perhaps useful to an algebraic geometer not familiar with the area, classical though it may be, then has their main argument, and concludes with the geometric applications concerning Torelli. The writing throughout is quite clean and well-organized and gives nice motivation and context for all the results and techniques they use.