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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 $X_1$ at 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 $n$.
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 $C$, $X_2$ is the blow-up of $X_1$ at 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 arbitrary $n$.