The generic isogeny decomposition of the Prym Variety of a cyclic branched covering

Abstract

Let f:S'longrightarrow S be a cyclic branched covering of smooth projective surfaces over {mathbb {C}} whose branch locus Delta subset S is a smooth ample divisor. Pick a very ample complete linear system |{mathcal {H}}| on S, such that the polarized surface (S, |{mathcal {H}}|) is not a scroll nor has rational hyperplane sections. For the general member [C]in |{mathcal {H}}| consider the mu _{n}-equivariant isogeny decomposition of the Prym variety {{,mathrm{Prym},}}(C'/C) of the induced covering f:C'{:}{=}f^{-1}(C)longrightarrow C: Prym(C′/C)∼∏d|n,d≠1Pd(C′/C).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {{\\,\\mathrm{Prym}\\,}}(C'/C)\\sim \\prod _{d|n,\\ d\\ne 1}{\\mathcal {P}}_{d}(C'/C). \\end{aligned}$$\\end{document}We show that for the very general member [C]in |{mathcal {H}}| the isogeny component {mathcal {P}}_{d}(C'/C) is mu _{d}-simple with {{,mathrm{End},}}_{mu _{d}}({mathcal {P}}_{d}(C'/C))cong {mathbb {Z}}[zeta _{d}]. In addition, for the non-ample case we reformulate the result by considering the identity component of the kernel of the map {mathcal {P}}_{d}(C'/C)subset {{,mathrm{Jac},}}(C')longrightarrow {{,mathrm{Alb},}}(S').