We consider geometrical problems on Gorenstein hypersurface orbifolds of dimension n ≥ 4 through the theory of Hilbert scheme of group orbits.

We give equivalent and sufficient criteria for the automorphism group of a complete toric variety, respectively a Gorenstein toric Fano variety, to be reductive. In particular we show that the automorphism group of a Gorenstein toric Fano variety is reductive, if the barycenter of the associated reflexive polytope is zero. Furthermore a sharp bound on the dimension of the reductive automorphism...

Let Q be a finite connected quiver and Λ the radical square zero algebra of Q. We give bijection between reduced differential projective modules over representations opposite If has oriented cycles is not basic cycle, we prove that dual numbers virtually Gorenstein.

We present an explicit method to produce upper bounds for the dimension of the moduli spaces of complete integral Gorenstein curves with prescribed symmetric Weierstrass semigroups.