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Let $(A, \mathfrak{m}, \kappa=A/\mathfrak{m})$ be a local ring and $f:X \to \operatorname{Spec} (A)$ a scheme. Let $D \subset X$ a divisor on $X$ contained in special fiber $D \subset f^{-1}(\sigma_{\mathfrak{m}})$ with associated ideal sheaf $I_D$.

Assume we know that $I_D$ is generated by a power of $\mathfrak{m}$, means there exist a $n$ with $I_D = \mathfrak{m}^nO_X$. The structure morphism induces natural morphism of sheaves $\mathfrak{m}^n \otimes_A \kappa \to I_D \otimes_{O_X} O_D$ and this induces a morphism on global sections $\mathfrak{m}^n \to H^0(D, I_D \otimes_{O_X} O_D)$.

What do we know about the last morphism? When is that an surjection?

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    $\begingroup$ So, you want your local ring $A$ of Krull dimension 1. And do you want $D$ to be Cartier, which probably means that $A$ should be a DVR? $\endgroup$
    – abx
    Commented Mar 27, 2021 at 6:16
  • $\begingroup$ @abx: No, I not require that $A$ should have dimension $1$, for example $A$ could be an affine surface and $X$ the blowup of the closed point of $A$. $\endgroup$
    – user267839
    Commented Mar 29, 2021 at 0:05

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