Counter-examples of high Clifford index to Prym-Torelli
type: Item
instance of: scholarly article
title: Counter-examples of high Clifford index to Prym-Torelli
zbMATH Open document ID: 1263.14036
DOI: 10.1090/S1056-3911-2012-00587-6
author: Elham Izadi
author: Herbert Lange
published in: Journal of Algebraic Geometry
publication date: 2012-09-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/1001.3610
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 14H51
zbMATH DE Number: 6086800
zbMATH Keywords: Prym varieties
zbMATH Keywords: Prym-Torelli map
zbMATH Keywords: tetragonal construction
zbMATH Keywords: multiples of the minimal cohomology class on abelian variety
zbMATH Keywords: correspondences
MaRDI profile type: Publication
arXiv ID: 1001.3610
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cites work: Recovering the Curve Data from a General Prym Variety
review text: The present article addresses two fundamental questions in the theory of abelian varieties: (1) determine the non-injectivity locus of the Prym map; (2) determine which multiples of the minimal class of a principally polarized abelian variety are representable by an irreducible curve.NEWLINENEWLINERegarding question (1), the authors generalize Donagi's tetragonal construction to certain 4-to-1 simply ramified covers of a smooth curve of any genus. As a consequence, they show the existence of curves \(X\) of arbitrarily large Clifford index such that the Prym map is not injective at any étale double cover of \(X\). The proof of this result provided by the authors is interesting in its own right because it gives explicit correspondences that induce the (generalized) tetragonal isomorphisms among the Pryms in question. In particular, this gives a new proof of Donagi's result.NEWLINENEWLINERegarding question (2), given a simply ramified \(n\)-sheeted cover of smooth projective curves \(X \to Y\) and an étale double cover \(\tilde{X} \to X\), the authors produce curves \(\tilde{C}_1\) and \(\tilde{C}_2\) with natural morphisms to the Prym \(P\) of the cover \(\tilde{X} \to X\). Moreover, the authors show that the image of \(\tilde{C}_i\) in \(P\) for \(i=1,2\) has the class NEWLINE\[NEWLINE2^{n-1}\frac{[\Theta_P]^{g_X-2}}{(g_X-2)!},NEWLINE\]NEWLINE where \(\Theta_P\) is the principal polarization of \(P\) and \(g_X\) is the genus of \(X\). Their proof proceeds by degenerating to the case when \(Y\) is an irreducible rational nodal curve.
reviewed by: Maxim Arap
description: scientific article; zbMATH DE number 6086800