# lift sections on a thickened curve

Let $X$ a curve over an algebraically closed field $k$ and $D$ a divisor on X.

Fix an integer $N$ and a closed point $x$ on $X$, we assume that $\deg(D)$ is big enough such that we have a surjective bundle map

$\mathcal{H}=H^{0}(X, \mathcal{O}(D))\rightarrow \mathcal{O}(D)\otimes_{\mathcal{O}_{X}} \mathcal{O}_{X,x}/\mathfrak{m}_{x}^{N+1}$

Now we consider an artinian ring $R$,$I$ an ideal of $R$, such that $I^2=0$, $\bar{R}:=R/I$ and $X_{R}$ the thickened curve.

Let a relative effective Cartier divisor $D\subset X_{R}$ defined by one equation $f$ such that it $(\bar{f})=\mathfrak{m}_{x}$ (the bar' is for the reduction). We denote by $A$ the ring of functions of the formal neighborhood of $D$, whch will be complete for the (f)-adic topology.

Now we take an element $b_{x}\in A/(f^{N+1})$ such that modulo $I$, it comes from an element $b_{0}\in \mathcal{H}(\bar{R})$, can we lift $b_{0}$ in $b\in \mathcal{H}(R)$ that maps to $b_{x}$?

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