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In the course of a proof, I need some version of Castelnuovo and Artin's contractibility criteria for a family of surfaces. Say I have a family (flat is probably needed, in order to compare intersection products as varying fibers) of surfaces $(\mathcal{X},\mathcal{B}) \rightarrow T$, where $\mathcal{B}_t \subset \mathcal{X}_t$ cuts either a $(-1)$-curve or some configuration of curves admitted by Artin's criterion for every fiber $t$. Assume that the configuration is independent of $t$. Then, I think some version of the contractibility criteria should hold. In particular, there should be an open subset $U \subset T$ such that I can perform Castelnuovo or Artin's contractions in family.

Just looking at the proof of the criteria for one fixed surface, it seems one can mimic the proofs to get a contraction in family (up to shrinking the base). My question are then the following.

Is this proved somewhere? If so, what is a reference?

If it is not proved, does it seems plausible? Are there some assumptions on the setup I should require in order for it to work?

If everything works as I expect, what is the obstruction (if any) to extend the contraction over the whole base $T$?

Edit

HYL's nice answer shows that what I am looking for is true in the category of complex manifolds. How about the algebraic one? I would like to claim I can get a scheme projective over $T $ (or what is left of it after shrinking).

Further edit

Doing some computations, Castelnuovo's criterion seems to work. On the other hand, extending Artin's criterion seems more subtle. Indeed, in the proof of the usual criterion, he needs to lift sections from a non reduced curve. Now, this may give some headache working in families, since the standard results that compare fibers (e.g. Hartshoner Chapter III.12) work well with irreducible fibers.

Is there maybe a version of Artin's criterion for excellent two dimensional schemes (or something similar)? This should work for the generic fiber, and then one could spread the contraction on an open neighborhood.

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  • $\begingroup$ I believe, if $\mathcal{X},\mathcal{B}$ are flat and projective over a reasonable (finite type over a field for example) $T$, eveyrthing should be alright. The proof should just follow the steps in Hartshorne. $\endgroup$ – Mohan Apr 22 '17 at 1:12
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    $\begingroup$ Depending on the situation (e.g. if it really just is a family of $(-1)$-curves), the contraction you're after might just be a step of the $\mathcal X/T$-MMP, in which case the image is automatically projective and has reasonable singularities. But of course this is not going to work for more exotic contractions, which aren't $K_{\mathcal X/T}$-negative. $\endgroup$ – user47305 Apr 22 '17 at 19:56
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For complex manifolds, simultaneous contractions in a family has been studied by Riemenschneider in the paper "Deformations of Rational Singularities and their Resolutions". Theorem 1 in that article says that given a family of complex manifolds $\mathcal{X} \to T$ such that for some fiber $X_t$, there exists a contraction $\nu_t : X_t \to Z_t$ to a variety with at worst isolated singularities, then up to shrinking $T$, the map $\nu_t$ extends to a contraction $\nu : \mathcal{X} \to \mathcal{Z}$ over $T$.

Let me also mention the following well-known result concerning deformations of maps (see Theorem 2.1 in this article). If $\nu : X \to Z$ is a surjective map such that $\nu_* \mathcal{O}_X = \mathcal{O}_Z$ and $R^1\nu_*\mathcal{O}_X = 0$, then a small deformation $\mathcal{X} \to T$ of $X$ induces a deformation $\mathcal{X} \to \mathcal{Z}$ of $\nu$ over the same base.

From either of the two results above, if the base $T$ in your question is smooth, then up to shrinking $T$ the family version of Castelnuovo and Artin's contractibility criteria follows easily: Since $\mathcal{B}$ is isomorphic to $\mathcal{B}_t \times T$ over $T$ (up to shrinking $T$) and since $\nu(\mathcal{B}_t)$ is a point by assumption, each fiber of $\mathcal{B} \to T$ is contracted by $\nu$ to a point. As the contractions of $(-1)$ and $A$-$D$-$E$-curves are unique, the restriction of $\nu$ to each fiber $\mathcal{X}_s$ contracts $\mathcal{B}_s$ with the same type of contraction.

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  • $\begingroup$ Thank you very much! Do you know whether I can do something similar algebraically? I would like to end up with a scheme projective over $T $ $\endgroup$ – Stefano Apr 22 '17 at 14:59

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