Let $k$ an algebraically closed field and $X$ be a connected projective curve (= a $1$-dimensional, proper $k$-scheme). consider a closed subscheme $Z = V(\mathcal{J})$ of $X$ defined by nilpotent sheaf of ideals $\mathcal{J} \subset O_X$. denote by $i:Z \to X$ the closed immersion.
in the following we will use a couple of notations:
let $\mathcal{N}_X \subset O_X$ the sheaf of nilpotent elements of $O_X$ and $N_X := \mathcal{N}_X(X)$ it's global sections;
resp $\mathcal{N}_Z \subset O_Z$ " nilpotents of $O_Z$ with $N_Z := \mathcal{N}_Z(Z)$.
We assume that $$\mathcal{J} \cdot \mathcal{N}_X =0$$
this implies $\mathcal{J}^2=0$ and $H^i(X,\mathcal{J} \cdot \mathcal{N}_X)=0$ for all $i \ge 0$. the short exact sequence
\begin{equation} 0 \to \mathcal{J} \to \mathcal{O}_X \to i_*\mathcal{O}_Z \to 0 \tag{1}\label{eq:0105star} \end{equation}
induces on long exact Cech cohomology sequence the map $\rho: O_X(X) \to O_Z(Z)$ sitting in
$$0 \to H^0(X,\mathcal{J}) \to \mathcal{O}_X(X) \to \mathcal{O}_Z(Z) \to H^1(X,\mathcal{J}) \to ... $$
Q: how to verify that the hypothesis $\mathcal{J} \mathcal{N}_X =0$ implies that $N_Z^2 \subset \rho(N_X)$?
I think that the key is to relate some sheaf sequences containing
$\mathcal{N}_Z^2$, $\mathcal{N}_X, \mathcal{N}_X\mathcal{J}= \mathcal{J}^2=0$ in a sophisticated way with (1) by an appropriate 2D lattice diagram and then release the global section functor to these sequences. the exactness of the Cech cohomology sequence says $im(\rho)=Ker(O_Z(Z) \to H^1(\mathcal{J},X))$. this observeation leads me to the suspicion that such diagram chaising argument could work. does anybody see the "right" sequences should be here combined? or maybe another way to solve it?