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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)

MSC:
14K05 Algebraic theory of abelian varieties
14H40 Jacobians, Prym varieties
14H10 Families, moduli of curves (algebraic)
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References:
[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
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