Differential symmetric signature in high dimension

TL;DR

This paper computes the differential symmetric signature for certain rings, revealing cases where it matches the F-signature and where it differs, thus advancing understanding of invariants in algebraic geometry.

Contribution

The authors compute the differential symmetric signature for invariant rings and hypersurface rings, showing it can coincide with or differ from the F-signature, providing new insights into ring invariants.

Findings

Differential symmetric signature equals 1/|G| for invariant rings under finite group actions.

For certain hypersurface rings with isolated singularities, the differential symmetric signature is 0.

The differential symmetric signature can differ from the F-signature, as demonstrated by the hypersurface example.

Abstract

We study the differential symmetric signature, an invariant of rings of finite type over a field, introduced in a previous work by the authors in an attempt to find a characteristic-free analogue of the F-signature. We compute the differential symmetric signature for invariant rings $k[x_1,\dots,x_n]^G$ where $G$ is a finite small subgroup of $\mathrm{Gl}(n,k)$ and for hypersurface rings $k[x_1,\dots,x_n]/(f)$ of dimension $\geq3$ with an isolated singularity. In the first case, we obtain the value $1/|G|$, which coincides with the F-signature and generalizes a previous result of the authors for the two-dimensional case. In the second case, following an argument by Bruns, we obtain the value $0$, providing an example of a ring where differential symmetric signature and F-signature are different.