The moduli space of matroids
Matthew Baker, Oliver Lorscheid

TL;DR
This paper develops a comprehensive algebraic framework for matroids, introducing moduli spaces over pastures, and connects these structures to classical matroid properties like regularity and binary-ness.
Contribution
It constructs moduli spaces for matroids over pastures and links these to classical matroid invariants, providing new algebraic insights and proofs.
Findings
Construction of the moduli space $Mat(r,E)$ for matroids over pastures.
Identification of the universal pasture $k_M$ and its role in classifying matroid structures.
Characterization of regular and binary matroids via their foundations and associated algebraic objects.
Abstract
In the first part of the paper, we clarify the connections between several algebraic objects appearing in matroid theory: both partial fields and hyperfields are fuzzy rings, fuzzy rings are tracts, and these relations are compatible with the respective matroid theories. Moreover, fuzzy rings are ordered blueprints and lie in the intersection of tracts with ordered blueprints; we call the objects of this intersection pastures. In the second part, we construct moduli spaces for matroids over pastures. We show that, for any non-empty finite set , the functor taking a pasture to the set of isomorphism classes of rank- -matroids on is representable by an ordered blue scheme , the moduli space of rank- matroids on . In the third part, we draw conclusions on matroid theory. A classical rank- matroid on corresponds to a -valued point…
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