On subcanonical Gorenstein varieties and apolarity

TL;DR

This paper characterizes certain subcanonical Gorenstein varieties within varieties of minimal degree using apolarity, showing they correspond to Fermat hypersurfaces, thus linking geometric properties with algebraic apolarity conditions.

Contribution

It provides a new characterization of subcanonical Gorenstein subvarieties via apolarity, specifically identifying those with Fermat hypersurfaces as apolar counterparts.

Findings

Subcanonical Gorenstein varieties are arithmetically Gorenstein.

Apolar hypersurfaces of these varieties are Fermat hypersurfaces.

Characterization links geometric properties with algebraic apolarity.

Abstract

Let $X$ be a codimension 1 subvariety of dimension $>1$ of a variety of minimal degree $Y$. If $X$ is subcanonical with Gorenstein canonical singularities admitting a crepant resolution, then $X$ is Arithmetically Gorenstein and we characterise such subvarieties $X$ of $Y$ via apolarity as those whose apolar hypersurfaces are Fermat.