Sur la démonstration de A. Weil du théorème de Torelli pour les courbes. (On the proof of A. Weil of the Torelli theorem for curves). (French) Zbl 0623.32017

Debarre, O.

Compos. Math. 58, 3-11 (1986).

Let C be a compact Riemann surface of genus \(g\geq 1\). Let \(J^ d(C)\) denote the complex manifold of isomorphism classes of line bundles of degree d on C. The Jacobian \(J^ 0(C)\) operates simply transitively on the \(J^ d(C)\). Let \(\theta\) denote the image of \(C^{g-1}\) in \(J^{g- 1}(C).\)
The author gives a simple geometric proof of the following theorem. Let \(g\geq 3\). Then, for \(a\neq 0\) in \(J^ 0(C)\), \(\theta\cap a(\theta)\) is irreducible unless (1) \(a\equiv p-q\) for some p,q\(\in C\) or (2) \(a\in \pi^*(J^ 0(E))\) for a map \(\pi\) : \(C\to E\) of degree 2, where E is of genus 1.
For \(g\geq 5\), this theorem is due to A. Weil. As is well-known, the theorem leads to a quick proof of Torelli’s theorem.
The author’s proof uses the fact that the singular locus of \(\theta\) has dimension \(\leq g-3\), but is otherwise elementary and self-contained. The proof also yields a description of the components of \(\theta\cap a(\theta)\) in all cases.

Reviewer: R.R.Simha

MSC:

32G15	Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30F99	Riemann surfaces
14H40	Jacobians, Prym varieties
14H10	Families, moduli of curves (algebraic)
14H15	Families, moduli of curves (analytic)

Keywords:

Torelli theorem for curves; compact Riemann surface; complex manifold of isomorphism classes of line bundles; Jacobian

References:

[1]	E. Arbarello , M. Cornalba , P. Griffiths , J. Harris : The geometry of algebraic curves , 1 (à paraître). · Zbl 0559.14017
[2]	C. Ciliberto : On a proof of Torelli’s Theorem, Ravello , Springer Lecture notes no. 997 (1983) 113-123. · Zbl 0513.14016
[3]	D. Mumford : Curves an their Jacobians , Ann. Arbor, The University of Michigan Press, 1978. · Zbl 0316.14010
[4]	A. Weil : Zum Beweis des Torellischen Satzes , Nachr. Akad. Wiss. Gôttingen Math. Phys. Kl. 2 (1957) 33-53. · Zbl 0079.37002