Suppose $C$ is a smooth projective curve over complex numbers. The singularities of the theta divisor $\Theta$ in $Pic^{g-1}(C)$ is described in the literature. It is $W^{1}_{g-1}=\{l\in Pic^{g-1}(C): h^0(l)\geq 2\}$. I could find in the literature the expected dimension results. Are the singularities of $W^1_{g-1}$ known.

Question: What is the (exp)dimension of $Sing(W^1_{g-1})$, for a generic curve ?

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    $\begingroup$ you should have a look at 'Geometry of Algebraic Curves' by Arbarello, Cornalba, Griffiths, and Harris. Very detailed information on this question is in the latter chapters. $\endgroup$
    – meh
    Jun 24 '19 at 20:17

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