Consider a polynomial $P\left(x_{1},\ldots,x_{N}\right)\in\mathbb{C}\left[x_{1},\ldots,x_{N}\right]$ in $N$ variables, and let $C$ be the projective/affine/whatever curve/hypersurface/whatever defined over $\mathbb{C}$ by $P\left(x_{1},\ldots,x_{N}\right)=0$. Everything I've seen so far has been abstract; I'm looking for information that is oriented toward actually computing things for given concrete examples. Note: I'm new to this—I'm an analyst, not an algebraist.
I know that to construct the Jacobian variety associated to $C$, one integrates a basis of global holomorphic differential forms over the contours of the curve's homology group. Is there a classical algorithm/method (i.e., a Calculus III or Complex Analysis exercise) for computing such a basis for a given curve? If so, what is it? An answer or a reference with worked examples for curves of genus $\geq2$ would suffice. Let me be clear: my hope is to be able to learn a method that I can then work through on my own to compute the basis of differential forms for particular curves that I come up with.
Thanks in advance.