Prime form on compact Riemann surfaces

Question

I am interested in studying the line bundle $O(\Delta)$ over a product $C \times C$, where $C$ is a (complex) projective curve of genus $g$ and $\Delta$ is the diagonal divisor. It seems to be well known that, if $C$ is smooth (so it is a riemann surface), the only section of that line bundle is given by the prime form $E(x,y)$, defined through theta functions (with characteristic) and a square root of the canonical bundle associated to $C$ (this can be found in Mumford's book or Fay's book on theta functions). My question is: is there any generalization of this prime form for singular curves? Where might I get some references about theta functions over singular curves?