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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

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$\Theta$ 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.

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