The volume intrinsic to a commutative graded algebra

TL;DR

This paper investigates the behavior of the canonical module in deformations of Artinian reductions of graded rings, providing a geometric interpretation of intersection numbers without relying on classical geometric structures.

Contribution

It introduces a novel perspective on the canonical module's behavior under deformations, constructing geometric intuition from algebraic data.

Findings

Behavior of the canonical module under deformations is characterized.

A new interpretation of intersection numbers without cycles is proposed.

Properties of the normalization in this context are explored.

Abstract

Recent works of the authors have demonstrated the usefulness of considering moduli spaces of Artinian reductions of a given ring when studying standard graded rings and their Lefschetz properties. This paper illuminates a key aspect of these works, the behaviour of the canonical module under deformations in this moduli space. We demonstrate that even when there is no natural geometry around, we can give a viewpoint that behaves like it, effectively constructing geometry out of nothing, giving interpretation to intersection numbers without cycles. Moreover, we explore some properties of this normalization.