Global complete intersection and globally generated ideal sheaves Let $X$ be a global complete intersection closed subscheme in $\mathbb{P}^n_{\mathbb{C}}$ for some $n$. Is the ideal sheaf $\mathcal{I}_X$ of $X$ in $\mathbb{P}^n_{\mathbb{C}}$ globally generated?
 A: No; in fact it won't even have global sections.  Take $X$ to be a hypersurface as the simplest example: if it is given by a section of $\mathcal{O}(d)$, then its ideal sheaf is $\mathcal{I}_X \cong \mathcal{O}(-d)$, which doesn't have any global sections.

Edit: I'll add more details for people who are not familiar with this.  Suppose that $X \subset Y$ is a hypersurface given by a section of some line bundle $\mathcal{L}$.  Let $\{U_i\}$ be an open cover of $Y$, and $g_{ij}$ the transition functions for $\mathcal{L}$ with respect to this cover.  Then if $f_i \in \mathcal{O}_{U_i}$ is a local function cutting out $X$, we have
$$
f_i = g_{ij} f_j ~\mathrm{on}~ U_i\cap U_j ~.
$$
The ideal sheaf $\mathcal{I}_X$ is the subsheaf of $\mathcal{O}_Y$ consisting of sections vanishing on $X$.  On $U_i$, such a section must be a multiple of $f_i$, i.e. $h_i f_i$ for some $h_i \in \mathcal{O}_{U_i}$.  On overlaps, we must have
$$
h_i f_i = h_j f_j ~,
$$
since these are sections of the trivial bundle $\mathcal{O}_Y$.  Therefore
$$
h_i = \frac{f_j}{f_i}h_j = \frac{1}{g_{ij}} h_j ~.
$$
So $\mathcal{I}_X \cong \mathcal{L}^{-1}$.
