Let k k be a field of characteristic zero containing a primitive fifth root of unity. Let X/k X/k be a smooth cubic threefold with an automorphism of order five, then we observe that over a finite extension of the field actually the dihedral group D 5 D5 is a subgroup of Aut(X) Aut(X) . We find that the intermediate Jacobian J(X) J(X) of X X is isogenous to the product of an elliptic curve E E and the self-product of an abelian surface B B with real multiplication by Q(5 – √ ) Q(5) . We give explicit models of some algebraic curves related to the construction of J(X) J(X) as a Prym variety. This includes a two parameter family of curves of genus 2 2 whose Jacobians are isogenous to the abelian surfaces mentioned as above.

On intermediate Jacobians of cubic threefolds admitting an automorphism of order five / T. Yamauchi, B. van Geemen. - In: PURE AND APPLIED MATHEMATICS QUARTERLY. - ISSN 1558-8599. - 12:1 Special Issue: In Honor of Eduard Looijenga(2016), pp. 141-164. [10.4310/PAMQ.2016.v12.n1.a5]

On intermediate Jacobians of cubic threefolds admitting an automorphism of order five

B. van Geemen
2016

Abstract

Let k k be a field of characteristic zero containing a primitive fifth root of unity. Let X/k X/k be a smooth cubic threefold with an automorphism of order five, then we observe that over a finite extension of the field actually the dihedral group D 5 D5 is a subgroup of Aut(X) Aut(X) . We find that the intermediate Jacobian J(X) J(X) of X X is isogenous to the product of an elliptic curve E E and the self-product of an abelian surface B B with real multiplication by Q(5 – √ ) Q(5) . We give explicit models of some algebraic curves related to the construction of J(X) J(X) as a Prym variety. This includes a two parameter family of curves of genus 2 2 whose Jacobians are isogenous to the abelian surfaces mentioned as above.
cubic threefolds; intermediate Jacobian; elliptic curves, and abelian surfaces with real multiplication
Settore MAT/03 - Geometria
PURE AND APPLIED MATHEMATICS QUARTERLY
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/476038
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