All the scientific output is contained in peer-reviewed articles mentioned in the specific section.
A more detailed description of the results can be found in Part B.
1) De Stefani and I proved that the second coefficient of the F-signature function exists for invariant rings under the action of a finite small group whose order is not divided by the characteristic of the field.
Moreover, using notions from representation theory we are able to prove that in this setting the F-signature function takes the shape of a quasi-polynomial.
We are also able to give a description of the other coefficients in terms of invariants of the finite acting group.
When the group is cyclic, we obtain more specific formulas for the coefficients of the quasi-polynomial, which allow us to compute the general form of the function in several examples of interest such as Veronese rings and Iyama-Yoshino’s singularities.
2) Brenner and I generalize our previous results on the differential symmetric signature to higher dimension.
We prove that the differential symmetric signature for invariant rings under the action of a finite small group G whose order is not divided by the characteristic of the field is equal to the value 1/|G|, which coincides with the F-signature.
We compute the differential symmetric signature for hypersurface rings of dimension ≥3 with an isolated singularity, obtaining the value 0.
While the first result extends a result of Watanabe and Yoshida from F-signature to the setting of differential symmetric signature, there is no analogue of the second one for F-signature.
Actually, we use it to exhibit an example of a ring where the differential symmetric signature and the F-signature are different.
3) Using the Gale transform and some inductive argument, Giansiracusa, Moon, Schaffler, and I construct polynomials whose associated variety W_{d,n} contains the Veronese compactification V_{d,n}.
Moreover, we prove that if d=2 or (d,n)=(3,7), (3,8), or (4,8) then W_{d,n}= V_{d,n}. In particular, in these cases we have equations that cut out V_{d,n}.
In addition, for n=d+4 or d=3 we are able to prove that W_{d,n} is given by the union of V_{d,n} and the determinantal variety that parametrizes configuration of points lying on a common hyperplane.
We also pinpoint several challenges involved in eliminating this extra component.
