Complex multiplication, Griffiths-Yukawa couplings, and rigidity for families of hypersurfaces
Let Md,n be the moduli stack of hypersurfaces X ⊂ Pn of degree d ≥ n + 1, and let M(1) d,n be the sub-stack, parameterizing hypersurfaces obtained as a d-fold cyclic covering of Pn−1 ramified over a hypersurface of degree d. Iterating this construction, one obtains M(ν) d,n. We show that M(1) d,n is rigid in Md,n, although for d < 2n the Griffiths-Yukawa coupling degenerates. However, for all d ≥ n + 1 the sub-stack M(2) d,n deforms. We calculate the exact length of the Griffiths-Yukawa coupling over M(ν) d,n, and we construct a 4-dimensional family of quintic hypersurfaces g :Z →T in P4, and a dense set of points t in T, such that g−1(t) has complex multiplication.
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