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Moduli of Abelian Varieties with Higher Degree Polarizations

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1958)

Fix integers g, d ≥ 1 and let A g,d denote the moduli stack associating to a scheme T the groupoid of pairs (A, λ), where A is an abelian scheme over T and λ : AAt is a polarization of degree d. Recall that this means that the kernel of λ is a finite flat group scheme over T of rank d2, and that fppf locally on T there exists an ample line bundle L on A such that the map

$${\rm{\lambda }}_L :A \to A^t ,a \to \left[ {t_a^* L \otimes L^{ - 1} } \right]$$

is equal to λ. In this case if f : AT is the structural morphism then f*L is locally free of rank d on T and its formation commutes with arbitrary base change on T (see for example [36, I, §1] for a summary of basic properties of ample line bundles on abelian varieties).

Keywords

  • Line Bundle
  • Abelian Variety
  • Group Scheme
  • Ample Line Bundle
  • Standard Construction

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© 2008 Springer-Verlag Berlin Heidelberg

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(2008). Moduli of Abelian Varieties with Higher Degree Polarizations. In: Compactifying Moduli Spaces for Abelian Varieties. Lecture Notes in Mathematics, vol 1958. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70519-2_6

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