Deformation of a generically finite map to a hypersurface embedding

TL;DR

This paper characterizes projective manifolds that deform into hypersurfaces within smooth projective manifolds, revealing their structure as iterated univariate coverings of normal type, with applications to projective space and Abelian varieties.

Contribution

It provides a structure theorem for manifolds deforming into hypersurfaces, introducing the concept of normal type univariate coverings, advancing the understanding of deformation theory in algebraic geometry.

Findings

Manifolds deforming into hypersurfaces are of special iterated univariate covering type.

The structure theorem applies to cases where the ambient space is projective space or an Abelian variety.

The concept of normal type coverings links line bundles to the normal bundle of the image.

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.