If C is generic of genus 6, then $\Theta_{sing}$ is a smooth surface. Can anyone give me a reference or a hint as to what that surface might be. What are the numerical characteristics of this surface? It has an involution. What is the quotient of this surface? How is this surface related to the curve C? I would ideally like information along the lines of the genus 5 case in which case $\Theta_{sing}$ is a double cover of a curve of genus 6 and the Prym of this cover is the original C. In case it helps, the inverse image of $\Theta_{sing}$ in $\Sym^5(C)$ is a 3 fold which is determinantal.
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$\begingroup$ You probably already know this, but for the benefit of others I point out that the nongeneric cases are described on page 211 of the book "Geometry of Algebraic Curves" by Arbarello, Cornalba, Griffiths and Harris. One can get some information about the surface, like the dimension of the cohomology groups of the structure sheaf, from these descriptions using degeneration and the semicontinuity theorem. $\endgroup$ – ulrich Jul 12 '15 at 12:14

$\begingroup$ @ulrich . I am travelling and don't have my copy of ACGH handy, but does it discuss genus 6 or genus 5 curves ? $\endgroup$ – aginensky Jul 12 '15 at 15:44

$\begingroup$ @ulrich p.211 does indeed discuss two cases where the surface isn't smooth. I think the case of a plane quintic is interesting. I have never been able to see how to go from that case to the generic case. If you have more thoughts on that, I would love to hear them. $\endgroup$ – aginensky Jul 12 '15 at 20:40

$\begingroup$ In the paper of Arbarello and Harris, on Canonical curves and quadrics of rank 4, pp 171174 (footnote), there is a description of this surface as an etale double cover of one of the 6 components of the surface of quadrics of rank 4 in P^5 containing the canonical model of the genus 6 curve C, if I understood correctly. The other 5 components are all 2planes. Maybe this will help. I.e. the curve C lies on a Veronese surface which lies on five 2 planes of rank 4 quadrics, and then there is another component of such quadrics through C not containing the Veronese. $\endgroup$ – roy smith Jul 13 '15 at 3:17

1$\begingroup$ I'm not sure if @roysmith's comment answered your question. In any case, for a plane quintic the surface $X$ is $C \times C$ with the diagonal blown down. Denoting the blow down morphism by $p$, one sees using the theorem on formal functions that $R^1 p_* \mathcal{O}_{C \times C}$ is is $7$dimensional ($g+1$ for general $g>1$). One also sees that $h^1(X, \mathcal{O}_X) = 6$ since line bundles on $X$ are line bundles on $C \times C$ trivial on the diagonal. $\endgroup$ – ulrich Jul 14 '15 at 5:45