Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 $n$.

share|improve this question
    
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 '12 at 15:55
1  
@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 '12 at 16:27
    
yes, I meant blow up of ${\mathbb P}^n$ along $C$, sorry about the imprecise wording –  Roman Jun 15 '12 at 17:25
    
wording corrected –  Roman Jun 15 '12 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 '12 at 0:38
show 1 more comment

1 Answer 1

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).

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.