Hurwitz spaces of triple coverings of elliptic curves and moduli spaces of abelian threefolds

Details Hurwitz spaces of triple coverings of elliptic curves and moduli spaces of abelian threefolds Hurwitz spaces of triple coverings of elliptic curves and moduli spaces of abelian threefolds

2002-07-31

arxiv.org/abs/math/0207305v3

Ann. Mat. Pura Appl. (4) Vol. 183 (2004), pp. 333 - 374.

Mathematics

Algebraic Geometry

43 pages, latex2e, some typos corrected, references updated, to appear in Annali di Matematica Pura ed Applicata

Scientific paper

10.1007/s10231-003-0098-9

We prove that the moduli spaces A_3(D) of polarized abelian threefolds with polarizations of types D=(1,1,2), (1,2,2), (1,1,3) or (1,3,3) are unirational. The result is based on the study of families of simple coverings of elliptic curves of degree 2 or 3 and on the study of the corresponding period mappings associated with holomorphic differentials with trace 0. In particular we prove the unirationality of the Hurwitz space H_{3,A}(Y) which parameterizes simply branched triple coverings of an elliptic curve Y with determinants of the Tschirnhausen modules isomorphic to A^{-1}.

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