Geometric interactions between bricks and $τ$-rigidity

Abstract: For finite dimensional algebras over algebraically closed fields, we prove some new results on the interactions between bricks and $\tau$-rigid modules, and also on their geometric counterparts-- brick components and generically $\tau$-reduced components. First, we give a new characterization of the locally representation-directed algebras and show that these are exactly the algebras for which every brick is $\tau$-rigid (hence, all indecomposable modules are bricks). We then consider analogous phenomena using some irreducible components of module varieties: the indecomposable components, brick components, and the indecomposable generically $\tau$-reduced components. For an algebra $A$, we prove that if two of these three sets of components coincide, then every indecomposable $\tau$-rigid $A$-module is a brick. For brick-infinite algebras, we also provide a new construction that yields, in some cases, a rational ray outside of the $\tau$-tilting fan (also called $g$-vector fan). For tame and $E$-tame algebras, we obtain new results on some open conjectures on $\tau$-tilting fans and bricks.