The point scheme of quadratic algebras that are Ore extensions of a twist of the polynomial ring

The object of study in this article is the family of quadratic algebras A on $$n+1$$ generators (where n ∈ Z , n ≥ 1 ) that have $$n(n+1)/2$$ defining relations, where A is an Ore extension of a twist, by an automorphism, of the polynomial ring on n generators. We prove that the scheme-theoretic zero locus, Γ A , of the defining relations of A is the graph of an automorphism of a subscheme, PA , of Pⁿ , so that Γ A is isomorphic to the scheme that represents the functor of point modules over A. In contrast to the standard method, if n ≥ 2 , for computing PA as the zero locus, in Pⁿ , of pmatrix n(n+1)/2 \\ n+1 pmatrix homogeneous polynomials of degree $$n+1$$ in $$n+1$$ variables, we prove that PA can be exhibited as the scheme-theoretic zero locus, in Pⁿ , of at most $$n(n-1)/2$$ homogeneous cubic polynomials in $$n+1$$ variables. Five examples are included that illustrate the new method.