The moduli space of matroids

Abstract

In [3], Nathan Bowler and the first author introduced a category of algebraic objects called tracts and defined the notion of (weak and strong) matroids over a tract. In the first part of the paper, we summarize and clarify the connections to other algebraic objects which have previously been used in connection with matroid theory. For example, we show that both partial fields and hyperfields are fuzzy rings, that fuzzy rings are tracts, and that these relations are compatible with previously introduced matroid theories. We also show that fuzzy rings are ordered blueprints in the sense of the second author. Thus fuzzy rings lie in the intersection of tracts with ordered blueprints; we call the objects of this intersection idylls.We then turn our attention to constructing moduli spaces for (strong) matroids over idylls. We show that, for any non-empty finite set E, the functor taking an idyll F to the set of isomorphism classes of rank-r strong F-matroids on E is representable by an ordered blue scheme Mat(r,E). We call Mat(r,E) the moduli space of rank-r matroids on E. The construction of Mat(r,E) requires some foundational work in the theory of ordered blue schemes; in particular, we provide an analogue for ordered blue schemes of the “Proj” construction in algebraic geometry, and we show that line bundles and their global sections control maps to projective spaces, much as in the usual theory of schemes.Idylls themselves are field objects in a larger category which we call F1±-algebras; roughly speaking, idylls are to F1±-algebras as hyperfields are to hyperrings. We define matroid bundles over ordered blue F1±-schemes and show that Mat(r,E) represents the functor taking an ordered blue F1±-scheme X to the set of isomorphism classes of rank-r (strong) matroid bundles on E over X. This characterizes Mat(r,E) up to (unique) isomorphism.Finally, we investigate various connections between the space Mat(r,E) and known constructions and results in matroid theory. For example, a classical rank-r matroid M on E corresponds to a morphism Spec(K)→Mat(r,E), where K (the “Krasner hyperfield”) is the final object in the category of idylls. The image of this morphism is a point of Mat(r,E) to which we can canonically attach a residue idyll kM, which we call the universal idyll of M. We show that morphisms from the universal idyll of M to an idyll F are canonically in bijection with strong F-matroid structures on M. Although there is no corresponding moduli space in the weak setting, we also define an analogous idyll kMw which classifies weak F-matroid structures on M. We show that the unit group of kMw can be canonically identified with the Tutte group of M, originally introduced by Dress and Wenzel. We also show that the sub-idyll kMf of kMw generated by “cross-ratios”, which we call the foundation of M, parametrizes rescaling classes of weak F-matroid structures on M, and its unit group coincides with the inner Tutte group of M. As sample applications of these considerations, we show that a matroid M is regular if and only if its foundation is the regular partial field (the initial object in the category of idylls), and a non-regular matroid M is binary if and only if its foundation is the field with two elements. From this, we deduce for example a new proof of the fact that a matroid is regular if and only if it is both binary and orientable.