# Is a node in a generically smooth family of nodal curves a rational singularity?

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!

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.
• 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? Apr 20, 2022 at 13:31
• 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 $X_y$). Apr 20, 2022 at 15:57
• 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. Apr 20, 2022 at 18:08