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This monograph studies decompositions of the Jacobian of a smooth projective curve, induced by the action of a finite group, into a product of abelian subvarieties. The authors give a general theorem on how to decompose the Jacobian which works in many cases and apply it for several groups, as for groups of small order and some series of groups. In many cases, these components are given by Prym varieties of pairs of subcovers. As a consequence, new proofs are obtained for the classical bigonal and trigonal constructions which have the advantage to generalize to more general situations. Several isogenies between Prym varieties also result.
List of contents
Introduction.- Preliminaries and basic results.- Finite covers of curves.- Covers of degree 2 and 3.- Covers of degree 4.- Some special groups and complete decomposabality.- Bibliography.- Index.
About the author
Herbert Lange is a retired professor at the university of Erlangen-Nuremberg. His main research interests are abelian varieties and vector bundles on algebraic curves. He is the coauthor of several books, among them the Grundlehren volume "Complex Abelian Varieties" and for the series Progress in Mathematics, "Complex Tori".
RUBÍ E. RODRÍGUEZ is Professor of Mathematics at Universidad de La Frontera and Director of the Geometry at the Frontier Research Center. Her research interests include moduli spaces of curves and abelian varieties, with special attention to the action of groups and algebras.
Report
"The Decompositions. The entire book is quite thorough, with enough background to serve as a standalone reference for several mini courses." (John T. Cullinan, Mathematical Reviews, May, 2024)
"The book is very well written, and gives a number of results, and of examples, interesting in some fields of Algebraic Geometry, specially those concerning algebraic curves, or equivalently, Riemann surfaces. Also, it serves to recall the work of Sevín Recillas Pishmish, whose untimely death prevented him from continuing working on these topics." (José Javier Etayo, zbMATH 1514.14001, 2023)
