Spinors on algebraic plane curves

Question

I'm interested in parameterizing spinors on Riemann surfaces. For my purposes, it's best to represent the Riemann surfaces as immersed in $\mathbb{C}P^2$, i.e. as algebraic plane curves. Apparently, mapping the Riemann surface to $\mathbb{R}^3$ via immersion is equivalent to equipping the RS with a spinor. My question: is that immersion algebraic? If so - or by some other method - how do I find an algebraic expression for a spinor over an algebraic plane curve?