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 blowup of ${\mathbb P}^n$ along $C$, $X_2$ is the blowup 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 $X_d$, $d=[(n3)/2]$? For a particular value of $n$ Bertram identified $X_d$ (or something similar) with the moduli space of semistable rank 2 vector bundles on $C$; but I am interested in the case of an arbitrary $n$.
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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 blowup 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). 

