On the (LC) conjecture

We prove the (LC) conjecture of Hochster and Huneke in a series of nontrivial cases. This has many facilities. Recently, Brenner and Caminata answered a numerical evidence due to Dao and Smirnov on the generalized Hilbert-Kunz functions. As a first application, we reprove this by a short argument. As a second application, we give a proof of another numerical evidence. Third, we answer a question posted by Vraciu.