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Consider a family of elliptic curves over the open unit disc $D\subset \mathbb{C}$ which degenerates to the nodal elliptic curve over the point $0$. I'd like to show that such a family is diffeomorphic to the Tate curve, that is, a neighbourhood of the nodal elliptic curve (the point at infinity) in the compactified moduli stack of elliptic curves.

One thing I was thinking about is to start with the trivial family over the sliced disc $D\setminus \{z: \text{Re}(z) \geq 0, \text{Im}(z) =0\}$. Then the Picard-Lefschetz formula specifies the monodromy if we want to fill in the fibre over $0$ with the nodal curve, and in particular it tells us how to glue this to get a nontrivial family over $D^*$.

So the question becomes, why is there only one way (up to diffeomorphism) to glue the nodal elliptic curve into this family? Or can anyone supply another way of approaching this problem?

Edit: I should also ask for this family to form a smooth surface, and specify that the central fibre should have precisely one node and one component.

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    $\begingroup$ One can pull back teh Tate curve by the map $z \to z^n$ and get a different family, so one has to be careful about what one is asking. One way to deal with this would be to demand that the family forms a smooth surface (and the special fiber has exactly one component and one node). $\endgroup$
    – Will Sawin
    Commented Jul 8, 2021 at 21:35
  • $\begingroup$ This is a very good point. I should ask for smoothness, and yes I have the curve with a single node and a single component in mind. $\endgroup$
    – EJAS
    Commented Jul 8, 2021 at 22:16
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    $\begingroup$ Crossposted from MSE. Please see here about crossposting: specifically, expected waiting periods and the need to link all crossposts together. $\endgroup$
    – KReiser
    Commented Jul 8, 2021 at 22:28

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