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Canonical polarizations of Picard schemes. (English) Zbl 0543.14015

Während im allgemeinen die Jacobischen von Kurven vom Geschlecht 2 und 3 in Produkte von elliptischen Kurven zerfallen, wird hier die Existenz von Kurven vom Geschlecht 2 und 3 (über \({\mathbb{C}})\) mit sogenannten elementaren Jacobischen gezeigt. Eine abelsche Varietät wird hier elementar genannt, wenn sie keine nicht-trivialen abelschen Untervarietäten enthält. Genauer: Sei R der Ring der ganzen Zahlen in \({\bar {\mathbb{Q}}}_ p\). Es gibt Kurven C über R vom Geschlecht 2 und 3, derart, daß \(Pic^ 0_ R(C)\) unendlich viele R-Automorphismen hat und die Fasern von \(Pic^ 0_ R(C)\) über Spec R elementare abelsche Varietäten sind. \(Pic^ 0_ R(C)\) besitzt unendlich viele kanonische Polarisierungen. Da es Einbettungen \(R\hookrightarrow {\mathbb{C}}\) gibt, erhält man Kurven der gewünschten Art über \({\mathbb{C}}\).
Reviewer: F.Ischebeck

MSC:

14H40 Jacobians, Prym varieties
14K30 Picard schemes, higher Jacobians
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