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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