Regular and rigid curves on some Calabi-Yau and general-type complete intersections

Let $X$ be either a general hypersurface of degree $n+1$ in $\mathbb P^n$ or a general $(2,n)$ complete intersection in $\mathbb P^{n+1}, n\geq 4$. We construct balanced rational curves on $X$ of all high enough degrees. If $n=3$ or $g=1$, we construct rigid curves of genus $g$ on $X$ of all high enough degrees. As an application we construct some rigid bundles on Calabi-Yau threefolds. In addition, we construct some low-degree balanced rational curves on hypersurfaces of degree $n + 2$ in $\mathbb P^n$.