In its modern formulation Torelli’s theorem asserts that if the Jacobians of two curves are isomorphic as polarized abelian varieties, then the two curves are isomorphic. There are now many proofs of this result. In its original formulation the theorem was in fact more general, and possibly because of this, Torelli’s proof was somewhat complicated. In this article the author shows that Torelli’s approach can be made to give a simple proof of the modern formulation of the theorem. If G and C are the two curves with polarized Jacobian (J(C), ), the idea is to show that some translate zG of G in J(C) is contained in exactly g−1 translates of the theta divisor , say by the points p1, · · · , pg−1. Let Cg be the g-fold symmetric product of C. Since the translate of by pi is the image of Cg−1 +pi Cg under the Abel-Jacobi map, if we lift zG birationally to a curve G+ in Cg, we then have G+ Cg−1 +pi, i = 1, · · · , g −1. Hence, G+ = C +p1 +· · · pg−1 in Cg, so that G+ and therefore G is isomorphic to C. This proof is characteristic-free and makes essential use of Poincar´e’s formula and Riemann’s singularity theorem.

A simple proof of the theorem of Torelli based on Torelli’s approach.

COLLINO, Alberto
1987-01-01

Abstract

In its modern formulation Torelli’s theorem asserts that if the Jacobians of two curves are isomorphic as polarized abelian varieties, then the two curves are isomorphic. There are now many proofs of this result. In its original formulation the theorem was in fact more general, and possibly because of this, Torelli’s proof was somewhat complicated. In this article the author shows that Torelli’s approach can be made to give a simple proof of the modern formulation of the theorem. If G and C are the two curves with polarized Jacobian (J(C), ), the idea is to show that some translate zG of G in J(C) is contained in exactly g−1 translates of the theta divisor , say by the points p1, · · · , pg−1. Let Cg be the g-fold symmetric product of C. Since the translate of by pi is the image of Cg−1 +pi Cg under the Abel-Jacobi map, if we lift zG birationally to a curve G+ in Cg, we then have G+ Cg−1 +pi, i = 1, · · · , g −1. Hence, G+ = C +p1 +· · · pg−1 in Cg, so that G+ and therefore G is isomorphic to C. This proof is characteristic-free and makes essential use of Poincar´e’s formula and Riemann’s singularity theorem.
1987
100
16
20
Algebraic curve; Torelli theorem; Riemann’s singularity theorem; Abel-Jacobi map.
Collino Alberto
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/109719
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