$p$-anisotropy on the moment curve for homology manifolds and cycles

TL;DR

This paper demonstrates that certain algebraic structures related to homology manifolds and cycles exhibit total p-anisotropy, with implications for their geometric realization on the moment curve.

Contribution

It establishes the total p-anisotropy of the Gorensteinification of face rings of cycles and shows parameters can be chosen geometrically without genericity assumptions.

Findings

Gorensteinification of face rings is totally p-anisotropic in characteristic p

Linear system of parameters can be realized with points on the moment curve

Parameters do not need to be chosen very generically

Abstract

We prove that the Gorensteinification of the face ring of a cycle is totally $p$-anisotropic in characteristic $p$. In other words, given an appropriate Artinian reduction, it contains no nonzero $p$-isotropic elements. Moreover, we prove that the linear system of parameters can be chosen corresponding to a geometric realization with points on the moment curve. In particular, this implies that the parameters do not have to be chosen very generically.