Let $C$ be a projective curve (scheme of pure dimension $1$). This induces a short exact sequence $$0 \to \mathcal{I}_C \to \mathcal{O}_{\mathbb{P}^n} \to i_*\mathcal{O}_C \to 0$$ for some $n$ such that $i:C \hookrightarrow \mathbb{P}^n$ is a closed immersion. Suppose that $i_*\mathcal{O}_C$ is $m$regular for some integer $m$. Can we express the regularity of $\mathcal{I}_C$ in terms of the integer $m$.
No. Take for $C$ a rational curve of degree $d$ in $\mathbb{P}^3$. Then $\mathcal{O}_C$ is 1regular, but $H^1(\mathbb{P}^3,\mathcal{I}_C(k))$ is zero iff the restriction $H^0(\mathbb{P}^3,\mathcal{O}_{\mathbb{P}}(k))\rightarrow H^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(dk))$ is surjective. For dimension reasons, this implies that the regularity of $\mathcal{I}_C$ becomes arbitrarily large when $d$ gets large. 

