The structure of the Prym map
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 \textit{C. H. Clemens}. Since then \textit{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 [\textit{R. Friedman} and \textit{R. Smith}, Invent. Math. 67, 473-490 (1982; Zbl 0506.14042)].
moduli space of unramified double covers of smooth curves
scientific article; zbMATH DE number 3855277
scientific article