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 $m$.
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No. Take for $C$ a rational curve of degree $d$ in $\mathbb{P}^3$. Then $\mathcal{O}_C$ is 1-regular, 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.
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$\begingroup$ Thank you very much for the answer. Would it help in any way if we add an assumption saying that the degree of $C$ is less than $m$? $\endgroup$ Commented Feb 9, 2014 at 17:29
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$\begingroup$ Yes, definitely. Assuming $C$ is reduced and irreducible, there is a simple lower bound on the regularity of $\mathcal{I}_C$ in terms of the degree, due to Gruson-Lazarsfeld-Peskine, Inv. math. 72 (1983). $\endgroup$– abxCommented Feb 9, 2014 at 18:02
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$\begingroup$ Thank you. I am more interested in the case when $C$ is not reduced or irreducible. $\endgroup$ Commented Feb 9, 2014 at 18:50
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$\begingroup$ I would guess there are also results in that direction, but I am not an expert. A search on MathSciNet (e.g. starting from the Gruson-Lazarsfeld-Peskine paper) might give you some ideas. $\endgroup$– abxCommented Feb 9, 2014 at 19:28