MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
Look at the book of Birkenhake-Lange "Complex abelian varieties", Chapters 10 and 11. – Francesco Polizzi Jan 13 '11 at 14:49
Volume 2 of Mumford's tata lectures on theta, chapter IIIa, is entitled "an elementary construction of hyperelliptic jacobians". – roy smith Jan 13 '11 at 18:31
Also, take a look at Milne's article on Jacobians in the book "Arithmetc geometry". – Donu Arapura Jan 13 '11 at 18:34
and this question:… – roy smith Jan 13 '11 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 [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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.