Polarisations of type \((1,2,\dots ,2)\) on abelian varieties
type: Item
instance of: scholarly article
title: Polarisations of type \((1,2,\dots ,2)\) on abelian varieties
zbMATH Open document ID: 0879.14020
DOI: 10.1215/S0012-7094-95-08007-7
author: S. Ramanan
author: D. S. Nagaraj
published in: Duke Mathematical Journal
publication date: 1996-05-01T00:00:00Z
review text: The algebraic equivalence class of any ample line bundle on an abelian variety \(A\) of dimension \(r\) determines an \(r\)-tuple \(\delta= (\delta_1, \delta_2, \dots, \delta_r)\) of positive integers with \(\delta_1 |\delta_2 |\cdots |\delta_r\). This is called the type of the line bundle. One might wish to understand at least the generic behaviour of the linear system of \(\tau\), where \(\tau\) is a line bundle of type \(\delta\) on an abelian variety. We are primarily interested here in the case \(\delta =(1,2, \dots, 2)\). First, in the case, there is a base locus consisting of \(2^{2(r-1)}\) points. The linear system gives a morphism of the variety \(\sigma A\) obtained by blowing up these points. Moreover, it is invariant under the natural involution \(\iota\) (upon a proper choice of the line bundle in the algebraic equivalence class). If \(r\geq 4\), we show that the linear system imbeds \(\sigma A/ \iota\) at least birationally. We prove this by considering abelian varieties arising from the following construction. Let \(C\) be a (smooth projective) curve and \(\widetilde C\) a 2-sheeted covering of \(C\) ramified at four points. Then the Prym variety \(P\) is an abelian variety with a polarisation of type \((1,2, \dots, 2)\). We study the rational map given rise to by this linear system by bringing into play the moduli of vector bundles on \(C\) of rank 2 and trivial determinant. In this case (assuming that \(C\) is of genus at least 4), we show that the variety \(\sigma (P)/ \iota\) is imbedded birationally but that there is actually a surface over which the mapping is \((2:1)\). Moreover, the data consisting of \(\widetilde C\), \(C\), etc. can be recovered from the geometry of this imbedding, thus providing a Torelli-type theorem for these Pryms. -- By an obvious degeneracy argument, this proves the result for generic abelian varieties claimed above. However, it is possible that the linear system behaves even better in the case of generic abelian varie…
Mathematics Subject Classification ID: 14K05
Mathematics Subject Classification ID: 14H40
zbMATH DE Number: 850804
zbMATH Keywords: nonprincipal polarisations
zbMATH Keywords: ample line bundle on an abelian variety
zbMATH Keywords: type of the line bundle
zbMATH Keywords: Prym variety
MaRDI profile type: Publication
cites work: The index of elliptic operators. III
cites work: Q4853298
cites work: Spectral curves and the generalised theta divisor.
cites work: An Infinitesimal Study of the Moduli of Hitchin Pairs
cites work: Classification of vector bundles of rank 2 on hyperelliptic curves
cites work: The tetragonal construction
cites work: The structure of the Prym map
cites work: Groupe de Picard des variétés de modules de fibrés semi-stable sur les courbes algébriques. (Picard groups of moduli varieties of semi- stable bundles on algebraic curves)
cites work: On the projective normality of complete linear series on an algebraic curve
cites work: On the equations defining Abelian varieties. I-III
cites work: Q4053768
cites work: Moduli of vector bundles on a compact Riemann surface
cites work: Q3855314
cites work: Q4735270
cites work: Moduli Space of Semistable Pairs on a Curve
cites work: Theta-Characteristics and Stable Vector Bundles
cites work: Q3476968
cites work: Ample Divisors on Abelian Surfaces
cites work: Q4112043
cites work: Q3697151
Recommended article: Polarizations on abelian varieties
Recommended article: Q3592546
Recommended article: POLARIZED MORPHISMS BETWEEN ABELIAN VARIETIES
Recommended article: Q4715868
Recommended article: Higher syzygies on general polarized Abelian varieties of type (1,⋯,1,d)$(1,\dots ,1,d)$
Recommended article: Polarization estimates for abelian varieties
Recommended article: Polarizations of Abelian Varieties Over Finite Fields via Canonical Liftings
Recommended article: On algebraic families of polarized abelian varieties
Recommended article: The dual polarization of an abelian variety
description: scientific article; zbMATH DE number 850804