# references for abelian schemes

Hi, I have a very basic question. I am looking for references explaining how to construct explicitily a Jacobian starting from a curve or examples of projective equations for an abelian scheme. I know a little bit the theory in general so I need examples to fix it, at least in the cases which are not too complicate( or when it is possible). Thank you

• Look at the book of Birkenhake-Lange "Complex abelian varieties", Chapters 10 and 11. Jan 13, 2011 at 14:49
• Volume 2 of Mumford's tata lectures on theta, chapter IIIa, is entitled "an elementary construction of hyperelliptic jacobians". Jan 13, 2011 at 18:31
• Also, take a look at Milne's article on Jacobians in the book "Arithmetc geometry". Jan 13, 2011 at 18:34
• and this question: mathoverflow.net/questions/51204/… Jan 13, 2011 at 18:46

The equations defining the Jacobian of a curve as a projective variety become very complicated as soon as the genus of the curve is bigger than 1. In the case of genus 2 curves, say $\mathcal{C}:y^2 = f(x)$, Grant  gives an explicit embedding in $\mathbb{P}^8$ and the defining equations when $\deg(f) = 5$ and Flynn  gives an explicit embedding in $\mathbb{P}^{15}$, the 72 (!) defining equations of the projective variety, and the biquadratic forms defining the addition law for when $\deg(f) = 6$ (see also Cassels and Flynn's book  for an "updated" version of Flynn's work in the early 90s among other things). For this reason, most computations with Jacobians use the Mumford representation of points in $\operatorname{Sym}^2(\mathcal{C})$ together with Cantor's algorithm for the addition law.