Rank varieties over the generic hypersurface I

TL;DR

This paper introduces rank varieties for modules over generic hypersurfaces, extending the concept of support varieties, and demonstrates their ability to realize any projective variety.

Contribution

It defines rank varieties using extension of scalars and shows their versatility in representing all projective varieties as rank varieties.

Findings

Every projective variety can be realized as a rank variety.

Rank varieties are defined via extension of scalars, differing from traditional support varieties.

Properties of rank varieties over generic hypersurfaces are investigated.

Abstract

To every local complete intersection ring one may associate a so-called generic hypersurface. In this paper we introduce rank varieties for modules and complexes over the generic hypersurface. The definition uses extension of scalars, rather than restriction of scalars which are used to define the conventional support varieties over a local complete intersection. We show that every projective variety can be realized as the rank variety of a finitely generated module over the generic hypersurface. We also investigate several properties of these rank varieties.