Prym varieties and the geodesic flow on \(\mathrm{SO}(n)\)
type: Item
instance of: scholarly article
title: Prym varieties and the geodesic flow on \(\mathrm{SO}(n)\)
zbMATH Open document ID: 0566.58028
DOI: 10.1007/BF01451409
author: Stefanos Pantazis
published in: Mathematische Annalen
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164057
review text: The bigonal construction, which associates to any (arbitrarily ramified) double cover of a hyperelliptic curve \(K\to^{\pi}K_ o\to^{\pi_ 0}\mathbb P^ 1\) a similar tower \(C\to C_ 0\to\mathbb P^ 1\) where the points of \(C\) are the different ways of lifting a fibre of \(\pi_ 0\) to a pair of points in \(K\), is studied. \textit{R. Donagi's} tetragonal construction [Bull. Am. Math. Soc., New Ser. 4, 181--185 (1981; Zbl 0491.14016)] is used to show that the Prym varieties \(\text{Prym}(K)\), \(\text{Prym}(C)\) are isogenous; a more detailed analysis proves that they are dual abelian varieties. The map linearizing the geodesic flow on \(\mathrm{SO}(n)\) is then studied and the abelian varieties where it is rendered injective are found. Because of the above results on the bigonal construction, this generalizes, (and gives a direct geometric proof of) earlier results by \textit{L. Haine} [Math. Ann. 263, 435--472 (1983; Zbl 0521.58042)] and \textit{M. Adler} and \textit{P. van Moerbeke} [Invent. Math. 67, 297--331 (1982; Zbl 0539.58012)] for the \(\mathrm{SO}(4)\) case.
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 53D25
Mathematics Subject Classification ID: 14K99
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 37C10
zbMATH DE Number: 3903614
zbMATH Keywords: bigonal construction
zbMATH Keywords: double cover of a hyperelliptic curve
zbMATH Keywords: Prym varieties
zbMATH Keywords: geodesic flow
zbMATH Keywords: abelian varieties
MaRDI profile type: Publication
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OpenAlex ID: W2318730560
description: scientific article; zbMATH DE number 3903614