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

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Apr 20, 2022 at 18:08	comment	added	aiz89		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 15:57	comment	added	Hacon		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 13:31	comment	added	stupid_question_bot		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?
S Apr 20, 2022 at 6:04	review	First answers
Apr 20, 2022 at 6:41
S Apr 20, 2022 at 6:04	history	answered	aiz89	CC BY-SA 4.0