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For a reduced curve $Z$, a general linear projection from $Z$ to $\mathbb{P}^2$ is a birational morphism to a plane curve $C$ of degree $d=d(Z)$. Thus, $$1-\chi(Z,\mathcal{O}_Z) \leq 1-\chi(C,\mathcal{O}_C) = d(d-2)/2.$$ In general, this is the best possible inequality, since some hypersurfaces of degree $r$ in $\mathbb{P}^n$ contain a $2$-plane, and thus contain plane curves $C$ of degree $d$.