The moduli space of matroids

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