Complex multiplication, Griffiths-Yukawa couplings, and rigidity for families of hypersurfaces

Details Complex multiplication, Griffiths-Yukawa couplings, and rigidity for families of hypersurfaces Complex multiplication, Griffiths-Yukawa couplings, and rigidity for families of hypersurfaces

2003-07-31

arxiv.org/abs/math/0307398v2

Mathematics

Algebraic Geometry

38 pages, amsLaTeX, second version: Correction of a wrong statment in Section 8; References updated, some typos corrected (inc

Scientific paper

Let M(d,n) be the moduli stack of hypersurfaces of degree d > n in the complex projective n-space, and let M(d,n;1) be the sub-stack, parameterizing hypersurfaces obtained as a d fold cyclic covering of the projective n-1 space, ramified over a hypersurface of degree d. Iterating this construction, one obtains M(d,n;r). We show that M(d,n;1) is rigid in M(d,n), although the Griffiths-Yukawa coupling degenerates for d<2n. On the other hand, for all d>n the sub-stack M(d,n;2) deforms. We calculate the exact length of the Griffiths-Yukawa coupling over M(d,n;r), and we construct a 4-dimensional family of quintic hypersurfaces, and a dense set of points in the base, where the fibres have complex multiplication.

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