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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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