Moduli and smooth specialization of hypersurfaces in a smooth projective variety

In the first part of the colloquium talk, we will introduce some theories about construction of moduli space of hypersurfaces in the projective space.  In the second part, we give a structure theorem for projective manifolds $W_0$ with the property of admitting a one parameter deformation where $W_t$ is a smooth hypersurface in a smooth projective variety $Z_t$. Their structure is the one of special iterated univariate coverings, which we call normal type. We give an application to the case where $Z_t$ is a projective space, respectively an abelian variety. We also give a characterizaton of smooth ample hypersurfaces in abelian varieties and describe an irreducible connected component of their moduli space. The second part is based on joint work with Fabrizio Catanese.