  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
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.
