Deformation of a generically finite map to a hypersurface embedding

Abstract

Motivated by the theory of Inoue-type varieties, we give a structure theorem for projective manifolds $W_0$ with the property of admitting a 1-parameter deformation where $W_t$ is a hypersurface in a projective smooth manifold $Z_t$. Their structure is the one of special iterated univariate coverings which we call of normal type, which essentially means that the line bundles where the univariate coverings live are tensor powers of the normal bundle to the image $X$ of $W_0$. We give applications to the case where $Z_t$ is projective space, respectively an Abelian variety.