# on relative divisors over artinian rings

Let $X$ a curve over $\mathbb{C}$, $D$ a divisor on $X$, $R$ a local artinian ring of residue field $\mathbb{C}$

Let $A=H^{0}(X_{R},\mathcal{O}(D_{R}))$ the scheme of sections over $Spec(R)$.

Let $H\subset X_{R}$ an effective divisor such that his reduction over $\mathbb{C}$, $\bar{H}=[x]$ for a point $x\in X$. Fix an integer $N\in \mathbb{N}$, If $\deg(D)$ is big enough, do we have a smooth surjective morphism

$A\rightarrow \mathcal{O}(D_{R})\otimes_{\mathcal{O}_{X_{R}}}\mathcal{O}_{X_{R}}/I_{H}^{n}$

?

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