Let $X$ be a projective, reduced scheme of pure dimension one (irreducible components have dimension one) over some field $k$. Let $X_1,\ldots,X_r$ be the irreducible components of $X$ and let $f_i: X_i \to X$ denote the corresponding closed immersions.
I was wondering when (for what kind of schemes resp. for what kind of intersection behaviour of the components) the following happens: The closed immersions $f_i$ are all flat.
My background for this is: I have $\mathcal{O}_X$-modules $\mathcal{F}$ that are embedded into $\mathcal{K}_X$, the sheaf of meromorphic functions on $X$, and want to restrict them to the $X_i$. I am trying to find reasonable assumptions on $\mathcal{F}$ such that its restriction to the components is again embedded in $\mathcal{K}_{X_i}$. Doing the local analysis I find that even ideal sheaves do not restrict to something embedded.
I am grateful for any kind of input or help.
A flat map locally of finite presentation is open, so asking for a closed immersion (which is certainly locally of finite presentation) to also be flat means that it should be both a closed and open map. This only occurs when the irreducible component is also a connected component. So your scheme must be a disjoint union of it's irreducible components.