Let $k$ be an algebraically closed field (of arbitrary characteristic), and $C$ a smooth projective curve over $k$, given by defining equations in projective space. I am looking for an algorithmic construction of the Jacobian variety of $C$.
To be more precise :
- I am just looking for an explicit construction method, not an efficient algorithm. Complexity issues are not relevant, as long as there is no brute-force search on an infinite set.
- The algorithm should at least yield explicit affine equations of open charts covering the Jacobian as well as the transition maps between them (or of course equations of the Jacobian in projective space).
- The construction should apply to any such curve, in any characteristic. There are lots of references concerning hyperelliptic curves or curves over $\mathbb{C}$, but these do not apply to the general case.
For the moment, I am only aware of the following references :
Anderson, Abeliants and their application to an elementary construction of Jacobians : this seems to answer the question but I cannot figure out the geometric intuition behind it, nor is it clear to me how to get actual equations for the projective variety constructed there.
Shepherd-Barron, Thomae's formulae for non-hyperelliptic curves and spinorial square roots of theta-constants on the moduli space of curves : the results of this paper require a restriction on the characteristic of the ground field.
