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
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1$\begingroup$ Look at the book of Birkenhake-Lange "Complex abelian varieties", Chapters 10 and 11. $\endgroup$– Francesco PolizziCommented Jan 13, 2011 at 14:49
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$\begingroup$ Volume 2 of Mumford's tata lectures on theta, chapter IIIa, is entitled "an elementary construction of hyperelliptic jacobians". $\endgroup$– roy smithCommented Jan 13, 2011 at 18:31
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$\begingroup$ Also, take a look at Milne's article on Jacobians in the book "Arithmetc geometry". $\endgroup$– Donu ArapuraCommented Jan 13, 2011 at 18:34
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$\begingroup$ and this question: mathoverflow.net/questions/51204/… $\endgroup$– roy smithCommented Jan 13, 2011 at 18:46
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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 [1] gives an explicit embedding in $\mathbb{P}^8$ and the defining equations when $\deg(f) = 5$ and Flynn [2] 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 [3] 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.
[1] Grant, D. Formal groups in genus two. J. Reine Angew. Math. 411 (1990), 96–121.
[2] Flynn, E. V. The Jacobian and formal group of a curve of genus 2 over an arbitrary ground field. Math. Proc. Cambridge Philos. Soc. 107 (1990), no. 3, 425–441.
[3] Cassels, J. W. S.; Flynn, E. V. Prolegomena to a middlebrow arithmetic of curves of genus 2. London Mathematical Society Lecture Note Series, 230. Cambridge University Press, Cambridge, 1996. xiv+219 pp. ISBN: 0-521-48370-0
