On the determinant bundles of abelian schemes

Vincent Maillot; Damian Rössler

doi:10.1112/S0010437X07003235

On the determinant bundles of abelian schemes

Part of:Cycles and subschemes Abelian varieties and schemes

Published online by Cambridge University Press: 01 March 2008

Vincent Maillot and

Damian Rössler

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Vincent Maillot: Affiliation:
Institut de Mathématiques de Jussieu, Université Paris 7 Denis Diderot, C.N.R.S., Case Postale 7012, 2 place Jussieu, F-75251 Paris cedex 05, France (email: vmaillot@math.jussieu.fr, dcr@math.jussieu.fr)
Damian Rössler: Affiliation:
Institut de Mathématiques de Jussieu, Université Paris 7 Denis Diderot, C.N.R.S., Case Postale 7012, 2 place Jussieu, F-75251 Paris cedex 05, France (email: vmaillot@math.jussieu.fr, dcr@math.jussieu.fr)

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Abstract

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Let $\pi :\mathcal {A}\rightarrow S$ be an abelian scheme over a scheme S which is quasi-projective over an affine noetherian scheme and let $\mathcal {L}$ be a symmetric, rigidified, relatively ample line bundle on $\mathcal {A}$ . We show that there is an isomorphism $\det (\pi _*{\cal L})^{\otimes 24}\simeq (\pi _*\omega _{{\cal A}}^{\vee })^{\otimes 12d}$ of line bundles on S, where d is the rank of the (locally free) sheaf $\pi _*{\cal L}$ . We also show that the numbers 24 and 12d are sharp in the following sense: if N>1 is a common divisor of 12 and 24, then there are data as above such that $\det (\pi _*{\cal L})^{\otimes (24/N)}\not \simeq (\pi _*\omega _{{\cal A}}^{\vee })^{\otimes (12d/N)}.$