# Singularity of Brill-Noether sub varieties of Picard varieties of smooth curves

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 ?

• 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.
– meh
Jun 24 '19 at 20:17