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 general curve of genus 5. Consider the Abel-Jacobi map from the 4-th symmetric product $C^{(4)}$ to $Pic^4(C)$. Its image is the theta divisor. Because a general curve of genus $5$ has a one parameter family of $g^1_4$, the Abel-Jacobi map contracts a surface $S\subset C^{(4)}$ to a curve $X\subset\Theta$ with fibers $\mathbb{P}^1$. This map is a small resolution of singularities of $\Theta$.($\Theta$ has ordinary double point along $X$ by Riemann singularity theorem.) The local picture looks like the following. $\Theta$ has local equation (up to crossing $\mathbb{A}^1$) $$xy+zw=0$$
If we blowing up the ideal $(x,z)$ we get a small resolution. (Think about the picture of the Atiyah flop). So my question is there any intrinsic description of a subvariety $Y$ in $\Theta$, such that we blow up $\Theta$ along $Y$ we will get $C^{(4)}$? If yes, what does the singular cohomology of $Y$ look like? I think $Y$ is a smooth 3-fold.

Thank you very much. Any input is very helpful.

Jie Wang

share|improve this question
    
$\Theta$ has dimension 4, so if $Y$ is a 3-fold it is a divisor in $\Theta$, hence it's blow up is isomorphic to the base space.... By the way, I think you have some exceptional fibers of the map over effective even theta-characteristic. –  IMeasy Jan 31 '13 at 15:42
    
Hi, I think \Theta is of dimension 4 but is singular. It is why this is a small resolution. The exceptional loci is not a divisor. You only see some extra stuff near the singulat loci of \Theta. –  Jie Wang Jan 31 '13 at 17:23
    
I am putting this in because you say 'any input is helpful'. As I recall, if you look in A-C-G-H, this is explained. The singularities of the theta divisor are a 'determinantal variety'. The construction is more general, it explains the algebraic structure of 'the set of line bundles with more sections than the general one'. Have a look. –  aginensky Jan 31 '13 at 20:44
    
Thanks. Could you please be more specific?Like which chapter of ACGH? –  Jie Wang Jan 31 '13 at 22:50
    
In fact, I do know the local equation of \Theta. That is given by xy+zw=0. I know locally if I blow up the ideal (x,z) I get a small resolution. I was just asking for the global picture. Is there any intrinsically defined variety in \Theta with its ideal locally look like (x,z). Thanks. –  Jie Wang Jan 31 '13 at 22:57

1 Answer 1

Since I posted my comment a year ago, I've learned the answer to this question. In 'Geometry of Algbebraic Curves' as a series of exercises one shows the following- i) A general genus 5 curve is the intersection of 3 smooth quadrics in $P^4$. ii- These quadrics form a $P^2$. The quadrics of rank < 5 are always of rank 4 and form a smooth quintic curve , call that D, in the $P^2$. iii- This is genus 6 and has an unramified double cover which is a curve of genus 11, call that $W$. iv- Then $\Theta_{sing} = W$. Further buried in the literature is the claim that Pr(W/D) is isomorphic to Jac(C). If you are interested contact me and I can send you a short pdf with the 'known' results. Note $dim(\Theta_{sing}) = g-4$ always for Jacobians. The resolution, which is $C_4$ is a $P^1$ bundle over $\Theta_{sing} $ and an isomorphism elsewhere.

share|improve this answer

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.