$\tau$-Tilting Finiteness of Non-distributive Algebras and their Module Varieties

We treat the $\tau$-tilting finiteness of those minimal representation-infinite (min-rep-infinite) algebras which are non-distributive. Building upon the new results of Bongartz, we fully determine which algebras in this family are $\tau$-tilting finite and which ones are not. This complements our previous work in which we carried out a similar analysis for the min-rep-infinite biserial algebras. Consequently, we obtain nontrivial explicit sufficient conditions for $\tau$-tilting infiniteness of a large family of algebras. This also produces concrete families of "minimal $\tau$-tilting infinite algebras"-- the modern counterpart of min-rep-infinite algebras, independently introduced by the author and Wang. We further use our results on the family of non-distributive algebras to establish a conjectural connection between the $\tau$-tilting theory and two geometric notions in the study of module varieties introduced by Chindris, Kinser and Weyman. We verify the conjectures for the algebras studied in this note: For the min-rep-infinite algebras which are non-distributive or biserial, we show that if $\Lambda$ has the dense orbit property, then it must be $\tau$-tilting finite. Moreover, we prove that such an algebra is Schur-representation-finite if and only if it is $\tau$-tilting finite. The latter result gives a categorical interpretation of Schur-representation-finiteness over this family of min-rep-infinite algebras.