## blow-ups of secant varieties

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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What does "$X_1$ is the blow-up of $C$" mean? Did you intend to say "the blow-up of $\mathbb P^1$ at $C$"? – Will Sawin Jun 15 at 15:55
@Will -- I think the OP means the blowing up of $\mathbb{P}^n$ along the ideal sheaf of the embedded curve $C$. – Jason Starr Jun 15 at 16:27
yes, I meant blow up of ${\mathbb P}^n$ along $C$, sorry about the imprecise wording – Roman Jun 15 at 17:25
wording corrected – Roman Jun 15 at 17:27
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. – aginensky Jun 16 at 0:38

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).

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Can you give an example when X_k isn't smooth? -- thanks

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I think it is poor form to answer your own question with a second question. – Jason Starr Jun 17 at 1:20
Dear Roman, You have multiple accounts, e.g. the account posting this answer is not the same as the one posting your question. This will make it hard to interact with people on the site. You should go to meta and ask for your accounts to be merged. Regards, – Emerton Jun 17 at 4:42