The OP clarified that the question is not merely about what is stated and proved in the article of Clemens; the OP would like to know what has been proved after the article of Clemens. There is important work by Lawrence Ein, Gianluca Pacienza, Claire Voisin, and Geng Xu. To the best of my knowledge, the current state of the art is the following theorem of Riedl and Woolf that applies in arbitrary characteristic (not merely characteristic $0$).

Eric Riedl and Matthew Woolf

Rational curves on complete intersections in positive characteristic

https://arxiv.org/pdf/1609.05958.pdf

**Theorem 1.2.** [Riedl and Woolf] For every field $K$, for every pair of integers $n,d\geq 1$ with $d\geq 2n-1$, every "very general" degree $d$ hypersurface in $\mathbb{P}^n_K$ contains no rational curves, i.e., every morphism from $\mathbb{P}^1$ to the hypersurface is constant.

To be precise, in the projective $K$-scheme $\mathbb{P} H^0(\mathbb{P}^n_K,\mathcal{O}(d)) \cong \mathbb{P}^{\binom{n+d}{n}-1}_K$ parameterizing degree $d$ hypersurfaces in $\mathbb{P}^n$, there exist countably many irreducible, Zariski closed subschemes $Z_i$ of codimension $\geq d-(2n-2)$ such that for every algebraically closed extension $k/K$ and for every $k$-point of $\mathbb{P} H^0(\mathbb{P}^n_K,\mathcal{O}(d))$ that is contained in no $Z_i$, every $k$-morphism from $\mathbb{P}^1_k$ to the associated hypersurface in $\mathbb{P}^n_k$ is constant.

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