zbMATH — the first resource for mathematics

A family of abelian surfaces and curves of genus four. (English) Zbl 0821.14027
Let \(X\) be an abelian surface and let \(L\) be a polarization of type \((1,3)\) on \(X\). The corresponding map \(\varphi_ L : X \to \mathbb{P}^ 2\) is a 6-fold covering and one has 4 isogenies \(f_ i(X,L) \to (Y_ i, P_ i)\), \(i = 1, \dots, 4\), onto principally polarized abelian surfaces; when \((Y_ i, P_ i)\) is the Jacobian of a curve \(H\) of genus 2, then \(C = f_ i^{-1} (H)\) is a smooth curve of genus 4 and \(C \to H\) is an étale cyclic 3-fold covering.
In this paper, the authors describe carefully the case \(X = E \times E\) \((E =\) elliptic curve) and \(L = {\mathcal O}_ X (E \times \{0\} + \{0\} \times E + A)\) where \(A\) is the antidiagonal. In particular, they find the equation of the ramification curve of \(\varphi_ L\) in terms of the \(j\)-invariant of \(E\) and describe when the resulting principally polarized surfaces \((Y_ i, P_ i)\) are Jacobians; in these cases, the covering curve \(C\) has \(\operatorname{Aut} (C) = S_ 3 \times S_ 3\) hence, varying the elliptic curve \(E\), the authors construct a 1-dimensional family of curves of genus 4, with automorphism group \(S_ 3 \times S_ 3\).
Reviewer: L.Chiantini (Roma)

14K05 Algebraic theory of abelian varieties
14H40 Jacobians, Prym varieties
14H10 Families, moduli of curves (algebraic)
Full Text: DOI EuDML
[1] Birkenhake, Ch., Lange H.: Moduli spaces of Abelian Surfaces with Isogeny · Zbl 0880.14021
[2] Lange, H., Birkenhake, Ch.: Complex Abelian Varieties, Grundlehren 302, Springer Verlag (1982). · Zbl 1056.14063
[3] Katsura, T.: Generalized Kummer surfaces and their unirationality in characteristicp, J. Fac. Sci Univ. Tokyo, 34 (1987) 1–41 · Zbl 0664.14023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.