# Deformation of stable curve with regular total space [duplicate]

Let $$k$$ be a field, let $$X/k$$ be a stable curve. Is it always possible to find a deformation $$\mathcal{X}/k[[t]]$$ such that $$\mathcal{X}$$ is regular?

(Sorry for the confusion, this is a duplication of one of my previous post....The answer to this question is yes, by Theorem B.2 in Brian Conrad’s Appendix to “Specialization of linear systems from curves to graphs” by Matthew Baker.)

## marked as duplicate by Qixiao, user44191, Dima Pasechnik, LSpice, Denis NardinJul 15 at 18:02

Yes for $$k$$ algebraically closed. First of all, for each singularity, we can choose a local deformation over $$k[[t]]$$ such that the total space around this singularity is regular. In some local coordinates it can be written as $$k[[x, y, t]] / (xy-t)$$. To get a global deformation, one uses the local-global principle (1.5) of Deligne--Mumford.