Prime form on compact Riemann surfaces

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?

• This is false for smooth curves of $g=0$. For singular curves, what genus are you using? Jun 27, 2019 at 16:55
• I am interested in any genus greater than 0, but from your question I can guess that there is no general solution. I am mainly interested in cuspidal and nodal weierstrass cubic curves, so arithmetic genus 1. I am also interested in genus 2. Jun 27, 2019 at 17:04
• What is your definition of $O(\Delta)$? $O(-\Delta)$ is well defined and usually $O(\Delta)$ is its dual. For singular varieties, this is not a line bundle. For plane curves, with this definition, the same argument works to say that if the arithmetic genus is greater than $0$, $O(\Delta)$ has only one non-zero section. Jun 27, 2019 at 18:06
• till now I have just considered plane curves ( I still didn't even move to the genus 2 case), what would it be the problem with not plane curves? Might you provide me an example? In any case, I know that the same argument works to say that $O(\Delta)$ has only one, non trivial, section. I would like to know wheather there is an explicit way to descrive it in terms of some known functions.(like theta functions) Jun 27, 2019 at 19:29
• I do not think there is a reasonable notion of theta functions (at least I am not sure) for singular curves. Usually, theta functions are defined for the Jacobian and those represent line bundles for smooth curves and then they are compact. Picard group of a nodal (or cuspidal) curve is not an abelian variety. So, I do not know. Jun 27, 2019 at 22:27