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We construct Grassmannian categories of infinite rank, providing an infinite analogue of the Grassmannian cluster categories introduced by Jensen, King, and Su. Each Grassmannian category of infinite rank is given as the category of graded maximal Cohen–Macaulay modules over a certain hypersurface singularity. We show that generically free modules of rank |$1$| in a Grassmannian category of infinite rank are in bijection with the Plücker coordinates in an appropriate Grassmannian cluster algebra of infinite rank. Moreover, this bijection is structure preserving, as it relates rigidity in the category to compatibility of Plücker coordinates. Along the way, we develop a combinatorial formula to compute the dimension of the |$\textrm {Ext}^{1}$| -spaces between any two generically free modules of rank |$1$| in the Grassmannian category of infinite rank. [ABSTRACT FROM AUTHOR]

Gross–Hacking–Keel–Kontsevich [ 13 ] discuss compactifications of cluster varieties from positive subsets in the real tropicalization of the mirror. To be more precise, let |${\mathfrak {D}}$| be the scattering diagram of a cluster variety |$V$| (of either type– |${\mathcal {A}}$| or |${\mathcal {X}}$|⁠), and let |$S$| be a closed subset of |$\left (V^\vee \right)^{\textrm {trop}} \left ({\mathbb {R}}\right)$| —the ambient space of |${\mathfrak {D}}$|⁠. The set |$S$| is positive if the theta functions corresponding to the integral points of |$S$| and its |${\mathbb {N}}$| -dilations define an |${\mathbb {N}}$| -graded subalgebra of |$\Gamma (V, \mathcal {O}_V){ [x]}$|⁠. In particular, a positive set |$S$| defines a compactification of |$V$| through a Proj construction applied to the corresponding |${\mathbb {N}}$| -graded algebra. In this paper, we give a natural convexity notion for subsets of |$\left (V^\vee \right)^{\textrm {trop}} \left ({\mathbb {R}}\right)$|⁠ , called broken line convexity , and show that a set is positive if and only if it is broken line convex. The combinatorial criterion of broken line convexity provides a tractable way to construct positive subsets of |$\left (V^\vee \right)^{\textrm {trop}} \left ({\mathbb {R}}\right)$| or to check positivity of a given subset. [ABSTRACT FROM AUTHOR]