Let $X$ be a hypersurface of degree $r$ in $\mathbb{P}^n$, and $Z\subset X$ be a closed subscheme of pure dim 1. Let $g(Z):=1-\chi(\mathcal{O}_Z)$ and $d(Z)$ be its degree. I'm wondering that is there any bound $g(Z)\leq F_{r,n}(d(Z))$ where $F_{r,n}(-)$ is a degree two polynomial?

When $Z$ is irreducible and reduced, I think this follows from some classical results of curves in $\mathbb{P}^n$. But what will happen when $Z$ is reducible, or even non-reduced?

The main example I consider is a degree five smooth hypersurface in $\mathbb{P}^4$.