  We give an explicit formula to count the number of geometric branches of a
curve in positive characteristic using the theory of tight closure. This
formula readily shows that the property of having a single geometric branch
characterizes $F$-nilpotent curves. Further, we show that a reduced, local
$F$-nilpotent ring has a single geometric branch; in particular, it is a
domain. Finally, we study inequalities of Frobenius test exponents along purely
inseparable ring extensions with applications to $F$-nilpotent affine semigroup
rings.
