Rational curves of degree 10 on a general quintic threefold

We prove the "strong form" of the Clemens conjecture in degree 10. Namely, on a general quintic threefold F in P-4, there are only, finitely, many smooth rational curves of degree 10, and each curve C is embedded in F with normal bundle Moreover, in degree 10, there are no singular, reduced, and irreducible rational curves, nor any reduced, reducible, and connected curves with rational components on F.