Let $C$ be a curve embedded in ${\mathbb P}^n$ by a full linear system (I am particularly interested in the case of an elliptic curve but it seems natural to ask the question in more generality). Let $S_k$ be the $k$-th secant variety, i.e. the union of all $k$ planes intersecting $C$ at (at least) $k+1$ points. Construct $X_k$ inductively: $X_1$ is the blow-up of ${\mathbb P}^n$ along $C$, $X_2$ is the blow-up of $X_1$ along the proper preimage of $S_1$ in $X_1$ etc. Is there a reasonable moduli interpretation for $X_i$, especially for the last one $X_d$, $d=[(n-3)/2]$? For a particular value of $n$ Bertram identified $X_d$ (or something similar) with the moduli space of semi-stable rank 2 vector bundles on $C$; but I am interested in the case of an arbitrary $n$.
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$\begingroup$ What does "$X_1$ is the blow-up of $C$" mean? Did you intend to say "the blow-up of $\mathbb P^1$ at $C$"? $\endgroup$– Will SawinCommented Jun 15, 2012 at 15:55
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1$\begingroup$ @Will -- I think the OP means the blowing up of $\mathbb{P}^n$ along the ideal sheaf of the embedded curve $C$. $\endgroup$– Jason StarrCommented Jun 15, 2012 at 16:27
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$\begingroup$ yes, I meant blow up of ${\mathbb P}^n$ along $C$, sorry about the imprecise wording $\endgroup$– RomanCommented Jun 15, 2012 at 17:25
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$\begingroup$ wording corrected $\endgroup$– RomanCommented Jun 15, 2012 at 17:27
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$\begingroup$ I think this is very hard. Bertram in his range is able to prove that $X_k$ is smooth away from $X_{k-1}$. This will not be true in general. $\endgroup$– mehCommented Jun 16, 2012 at 0:38
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You can probaly describe your space as some kind of fibration in $\mathcal{M}_{0,m}$ over the dual linear system $\mathbb{P}^{n*}$, via Kapranov's blow-up construction of $\overline{\mathcal{M}_{0,m}}$ . This is done in
http://arxiv.org/abs/0903.5515
for some of the Bertram's cases but it should hold in general. Your space may have an interpretation in terms of moduli of vector bunldes (or sheaves).
