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The structure of the Prym map. (English) Zbl 0538.14019
The paper under review is a detailed treatment of the result, announced by the authors in Journées géométrie algébrique, Angers 1979, 143-155 (1980; Zbl 0464.14015) that the degree of the Prym map \(P: {\mathcal R}_ 6\to {\mathcal A}_ 5\) is 27, where \({\mathcal R}_ g\) (for \(g=6)\) is a moduli space of unramified double covers of smooth genus g curves. The main idea is to investigate the fiber over a general Jacobian, which can be found explicitly. To compute the degree near a non-zerodimensional fiber the authors use the infinitesimal investigation, which was generously outlined for them by C. H. Clemens. Since then V. Kanev [Izv. Akad. Nauk SSSR, Ser. Mat. 46, 244-268 (1982)] has improved these methods and established that the degree of the Prym map \(P: {\mathcal R}_ g\to {\mathcal A}_{g-1}\) is one if \(g=9\); for \(g=7\) and 8 it is also one [R. Friedman and R. Smith, Invent. Math. 67, 473-490 (1982; Zbl 0506.14042)].
Reviewer: V.V.Shokurov

MSC:
14H15 Families, moduli of curves (analytic)
14H40 Jacobians, Prym varieties
14B25 Local structure of morphisms in algebraic geometry: étale, flat, etc.
14K30 Picard schemes, higher Jacobians
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