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Hi Everybody,

I stumbled upon the following question for families of vector bundles (over a curve) but I guess it could be interesting to answer in general.

Suppose I have $B$ the blow-up of a smooth projective variety $M$ along a subvariety $N$. Let $E$ be the exceptional divisor over $N$. Suppose that $E$ has a section (or better that the normal bundle of $N$ has a nonvanishing section), then we can identify $N$ with the image of such section.

Then, suppose I have two flat families of objects: one $\Xi$ over $M$ and the other $\Psi$ over $E$, and suppose that they agree on $N$. Under what condition there exists a universal flat family $\Phi$ over $B$, whose restriction to $M$ (resp. to $E$) is equal to $\Xi$ (resp. to $\Psi$)?

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    $\begingroup$ Dear MBeasy, how is $N$ the "zero section" of $E$? The projection $E\to N$ has no sections in general, you need a nowhere vanishing section of the normal bundle to $N$ in $M$. $\endgroup$
    – Piotr Achinger
    Commented Jan 19, 2013 at 17:15
  • $\begingroup$ @Piotr: Yes you're right. I had the picture in my mind and I forgot to add the hypothesis. I'll edit this. $\endgroup$
    – IMeasy
    Commented Jan 19, 2013 at 17:34

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I think the answer is "basically never". Let $B$ be the simplest kind of blowup, that is, let $N$ be a single smooth point. Take a map from $M$ to some Hilbert scheme and a map from $E$ to some Hilbert scheme that agree on $N$. This gives two projective flat families that agree on $N$. To glue the families together to a projective flat family you would need to glue the maps together, impossible unless the map on $E$ is constant. To preserve this even for an arbitrary flat family, you can compose the maps with the maps to some nontrivial moduli space, so if you took the Hilbert scheme of cubic curves in $\mathbb P^2$, you would map to the compactified moduli space of elliptic curves. Then, again, to glue the maps together, you would need the map on $E$ to be a constant.

I can't imagine a niceness condition that would rule out this sort of example.

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  • $\begingroup$ Thank you, Will. But there must be some kind of conditions, because ecamples exist. For instance, I don't think that any family of curves parametrized by $\mathbb{F}_1$ is constant along the $(-1)$-curve. $\endgroup$
    – IMeasy
    Commented Jan 19, 2013 at 21:55
  • $\begingroup$ What's $\mathbb F_1$ in this context? $\endgroup$
    – Will Sawin
    Commented Jan 19, 2013 at 22:17
  • $\begingroup$ In fact, in my own case, the family over $E$ is given by strictly semistable bundles. The whole family is sent to one S-equivalence class in the semi-stable boundary of the moduli space, so it is constant (even if the moduli space is not fine). So this may well be one of the "constant" cases that are outisde of the hypotheses of your answer. $\endgroup$
    – IMeasy
    Commented Jan 19, 2013 at 22:31
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    $\begingroup$ Well I haven't studied this sort of thing so I don't fully understand your description of this moduli space, but I think the argument to prove whatever it is that's true goes like this: Take a line on $B$ that passes through some point on $E$. Look at its image in $M$. We can define two families on it - the restriction of the glued bundle from $E$, and the pullback from $M$. These families are the same except at one point. So the key question is, how different can they be at that point? For projective flat families, for instance, they must be the same. $\endgroup$
    – Will Sawin
    Commented Jan 19, 2013 at 22:52
  • $\begingroup$ I guess in your case you know the fibers are isomorphic, but the question is whether the isomorphism is natural? $\endgroup$
    – Will Sawin
    Commented Jan 19, 2013 at 22:53

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