  We prove the (LC) conjecture of Hochster and Huneke in some non-trivial
cases. This has several applications. Recently, Brenner and Caminata answered a
numerical evidence due to Dao and Smirnov on the shape of generalized
Hilbert-Kunz functions of smooth curves. As applications, we first reprove this
by a short argument. Then we give a proof of second numerical evidence
predicted by Dao and Smirnov on the shape of generalized Hilbert-Kunz functions
of nodal curves. Thirdly, we answer a question posted by Vraciu on the (LC)
property of a proposed ring. Inspiring with the (LC) property, we present a
connection to the stability theory. This leads us to investigate the stability
and the strong semistability of the sheaf of relations on $\{x^2,y^2,z^2\}$
over the Klein's quartic curve. This answers questions of Brenner. After
presenting a connection from (LC) to the $F$-threshold, we answer a question
posted by Huneke et al. Additional applications and examples are given.
