On the symmetric products of a curve
A celebrated theorem of Torelli asserts that two Riemann surfaces \(C\) and \(C'\) of genus \(g\) are isomorphic if and only if their \((g-1)\)-fold symmetric product \(C_{g-1}\) and \(C_{g-1}'\) are birationally equivalent. This theorem has been extended by \textit{H. H. Martens} [cf. Am. J. Math. 87, 257-261 (1965; Zbl 0137.405)] who proved that \(C\) and \(C'\) are isomorphic if and only if there is an integer \(n\), with \(1\leq n\leq g-1\), such that \(C_ n\) and \(C_ n'\) are birationally equivalent. The theorem cannot be extended in this form for \(n\geq g\). However in this paper we prove that if \(g\geq 3\), then \(C\) and \(C'\) are isomorphic if and only if there is an integer \(n\geq g\) such that \(C_ n\) and \(C_ n'\) are isomorphic. The proof is based on a Chern class computation and uses the well known criterion of Matsusaka-Ran and a theorem of Collino which describes the Chow ring of \(C_ n\).
scientific article; zbMATH DE number 446394