On the symmetric products of a curve
type: Item
instance of: scholarly article
title: On the symmetric products of a curve
zbMATH Open document ID: 0778.14013
DOI: 10.1007/BF01198726
author: Ciro Ciliberto
author: Edoardo Sernesi
published in: Archiv der Mathematik
publication date: 1993-11-15T00:00:00Z
review text: 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 \(C_ n\).
reviewed by: Ciro Ciliberto
Mathematics Subject Classification ID: 14H55
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 14H45
zbMATH DE Number: 446394
zbMATH Keywords: Torelli theorem
zbMATH Keywords: Riemann surfaces
zbMATH Keywords: symmetric product
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01198726
OpenAlex ID: W1994585840
description: scientific article; zbMATH DE number 446394