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Let $Y$ be a smooth irreducible curve over an algebraically closed field $k$ and $f : X\rightarrow Y$ be a proper flat morphism of relative dimension 1 with smooth generic fiber and whose closed fibers have at worst ordinary double points as singularities. If $x\in X$ is a node in the fiber over $f(x)$, then the complete local ring at $x$ is isomorphic to $k[[x,y,t]]/(xy-ut^e)$ for some unit $u\in k[[t]]$.

Is this a rational singularity? Are there other singularity types that such singularities are examples of?

This is a very naive question, so references and general information would be appreciated!

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This is more of a comment than an answer, but I don't have enough reputation to comment, so I'll post it as an answer.

First, I'm not sure the question makes sense as posted: your explicit complete local ring is of dimension 2, so it looks like you're assuming the base is a curve? Second, you need to assume that the base is smooth, or not too singular, if you want to be able say anything about the type of singularity at a node.

But yes, if $Y$ is a smooth curve, then the singularity at the node is what's known as an $A_n$-singularity (simplest type of Du Val singularity), and it is rational (the exceptional divisor of the resolution is a chain of rational curves).

Also, you can make statements of the type "if the singularities of $Y$ have property [...], then the singularities of $X$ have property [...]", even for $Y$ of higher dimension. If I remember correctly, though I'm less sure about this, the sharpest (in some sense) result of this type is with [...] = toric singularities.

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  • $\begingroup$ Thanks for your answer! This is what I suspected. (I'm also fine to assume that $Y$ is a smooth curve). Would you happen to have references for any of this? $\endgroup$
    – stupid_question_bot
    Commented Apr 20, 2022 at 13:31
  • $\begingroup$ As aiz89 points out there are many results in the litterature that show that if the singularities of a fiber are mild then so are those of the total space (on a neighborhood of the fiber). Typically these are known as "inversion of adjunction" results. So if $y\in Y$ is a smooth curve and $K_X$ is $\mathbb Q$-Cartier, then if the fiber $X_y$ is klt (resp. lc) so is $X$ (on a neighborhood of $X_y$). $\endgroup$
    – Hacon
    Commented Apr 20, 2022 at 15:57
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    $\begingroup$ I'm not sure about references, the $Y$ smooth curve case is very well-known. That it's an $A_{n-1}$ singularity ($xy=t^n$) follows from the fact that $xy=t$ over the line with coordinate t is the "versal deformation space of a node" (there is an explicit way to write down versal deformation spaces for any plane singularity, it's stated e.g. in Harris Morrison, Moduli of curves, pages 97-98; I assume Lectures on Deformations of Singularities by Michael Artin is an overkill, I've never read it). That the exceptional divisor is a chain of rational curves is e.g. in the Shafarevich textbook 4.3. $\endgroup$
    – aiz89
    Commented Apr 20, 2022 at 18:08

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