TITLE: blow-ups of secant varieties
QUESTION [8 upvotes]: 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 $n$.

REPLY [2 votes]: 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).