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Let $C$ be a curve (smooth projective over a field k) and $p$ a rational point on $C$. Put $\dot{C}=C-\{p\}$. Set $T=Spec R$ where $R$ is a noetherian k-algebra. Let $Y$ be a locally trivial fiber bundle (suppose for etale topology), with smooth affine fibers, over $\dot{C}_T:=\dot{C}\times T$. My question is as follows: can one extend the fiber bundle Y to a flat family over entire relative curve $C_T$?

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In this generality, it seems unlikely that anything can be said. What happens when $T$ is a point and $Y$ is a principal $G$-bundle, for an algebraic group $G$? – Keerthi Madapusi Pera Dec 11 '10 at 15:59
What do you mean by a flat family? You need to assume more than flatness (otherwise you could "extend" by taking the composite $Y \to \dot C \to C$). – Angelo Dec 11 '10 at 16:14
Right, I meant to extend the family over the point $p$, i.e. to add the fiber over $p$ in such a way that doesn't violate flatness of family, let's say "faithfully flat". As Keerthi Madapusi Sampath suggested, let us stick to the simplest case when $T$ is a point, and suppose further that we can embed the family $Y$ over $\dot{C}$ to some ambient space $E$ that's flat over $C$ (namely the case of $G$-bundles) such that $Y$ is closed inside the restriction of $E$ to $\dot{C}$, then the statement is implied by the properness of quot scheme. – Samuel Dec 11 '10 at 18:52
Dear Samuel, the Quot scheme exists when $E$ is proper over $C$. For proper maps, you just require that the extension be proper and flat, and that's it. In the affine case it is not at all clear to me what you should require. – Angelo Dec 12 '10 at 4:04

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