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Let $X\to \Delta$ be a flat family of complex surfaces with at most a finite number of singularities of simple type, where $\Delta$ is a complex domain in $\mathbb C$.

Here simple type means rational double point.

By a result of Tyurina, it is know that such deformations admit locally simultaneous resolutions after passing to a sufficiently high ramified cover of the base.

Conversely if $Y\to\Delta$ is a smooth family of complex surfaces and the central fiber $Y_0$ is the minimal resolution $Y_0\to X_0$ of a complex surface with a finite number of simple singularities, can we say that $Y\to \Delta$ must be (locally) a simultaneous resolution of some flat deformation of $X_0$ ?

I guess this boils down to ask whether it is possible to contract the exceptional curves of $Y_0$ within the family $Y\to\Delta$...

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up vote 2 down vote accepted

I think that the paper Burns-Wahl "Local contributions... " gives some of the examples you are looking for.

(sorry I just wanted to add a comment above, but I am not sure how to do it).

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If I understand your question, the answer is no.

You can consider any surface $X_0$ which does not admit a global smoothing. Let $Y_0\to X_0$ be the minimal resolution and consider a family $Y\to \Delta$ such that the general fiber does not admit any curve with negative self-intersection.

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Thanks, but how do you know that such a Y exists ? – Yann Jun 7 '11 at 6:24

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