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consider $S=Spec(\mathbb{C}[t])$ and $C\rightarrow S$ a family of proper curves with $C_{\mathbb{C}[t,t^{-1}]}$ smooth and $C_{t=0}$ nodal given by 2 irreducible components $C_1,C_2$ that intersect transversally at a point $p$. Assume that $\widehat{\mathcal{O}}_{C,p}\cong \mathbb{C}[[x,y]]/(xy-t)$. Let $\mathfrak{C}$ (resp. $\mathfrak{C_1}$, resp. $\mathfrak{C_2}$) be the formal completion of $C$ (resp. $C_1$,resp. $C_2$) at $t=0$ and $\mathfrak{M}$ be the sheaf of meromorphic functions of $\mathcal{O}_{\mathfrak{C}}$. Consider a function $f_1\in\mathcal{O}_{\mathfrak{C}_1}$ (resp. $f_2\in\mathcal{O}_{\mathfrak{C}_2}$) which restricts to a local generator of the sheaf $\mathcal{O}_{C_1}(-p)$ (resp. $\mathcal{O}_{C_2}(-p)$) at $p$. In my understanding this morally means that it corresponds to the coordinate $x$ (resp. $y$). Consider the formal subsheaf $\mathfrak{I}_t\subset \mathfrak{M}$ (resp. $\mathfrak{I}_p\subset \mathfrak{M}$) obtained by glueing $f_1\cdot\mathcal{O}_{\mathfrak{C}_1}$ and $f_2\cdot\mathcal{O}_{\mathfrak{C}_2}$ according to the rule

$f_1= t f_2$ resp. $f_1 = \frac{f_1}{f_2}f_2$

1 question: are the sheaves $\mathfrak{I}_p$ and $\mathfrak{I}_t$ coherent $\mathcal{O}_{\mathfrak{C}}$-modules? are they formal line bundles?

if 1 is true then by Grothendieck's theory they algebraize to coherent sheaves $I_t$ and $I_p$ on $C$. Since in the case of $I_p$ I am sending one coordinate to the other then 2 question: is it true that $I_p|_{t=0}\cong \mathfrak{m}_{C|_{t=0},p}$

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