Classification of noncommutative central conics

Classification of noncommutative quadric hypersurfaces is one of the major projects in noncommutative algebraic geometry. In recent years, we are dedicated to complete the classification of noncommutative central conics. To achieve this goal, we and other authors develop some theories to study and classify some classes of noncommutative quadric hypersurfaces in a series of papers. Finally, in this paper, we completely classify noncommutative central conics by developing the general theory of homogenization and dehomogenization for noncommutative algebras and by previous results. As a main result, we show that there are bijections among the following sets of objects (i) the set of isomorphism classes of $4$-dimensional Frobenius algebras, (ii) the set of isomorphism classes of noncommutative affine pencils of conics, and (iii) the set of isomorphism classes of noncommutative central conics.

💡 Research Summary

The paper “Classification of Noncommutative Central Conics” achieves a complete classification of noncommutative central conics, a long‑standing problem in noncommutative algebraic geometry. The authors develop a systematic theory of noncommutative homogenization and dehomogenization, introduce the notion of a strongly regular normal sequence, and use these tools to bridge three seemingly disparate families of objects: (i) 4‑dimensional Frobenius algebras, (ii) noncommutative affine pencils of conics, and (iii) noncommutative central conics.

The first part of the work revisits the foundations of quasi‑schemes, regular and normal elements, and AS‑Gorenstein algebras, establishing the language needed for later sections. A key technical obstacle is that homogenization of an algebra R depends on the chosen presentation of R, while dehomogenization of a graded algebra A depends on the choice of a regular normal element of degree 1. To overcome this, the authors define a “strongly regular normal sequence” and prove that, under this condition, the homogenization Hₙ(R) is independent of the presentation and the dehomogenization D_w(A) is independent of the chosen regular normal element (Theorem 5.1, Lemma 3.10). This yields well‑behaved functors between the categories of graded algebras and their non‑graded counterparts.

Next, the paper focuses on 3‑dimensional quantum polynomial algebras S and homogeneous central elements f∈Z(S)₂. The quotient A=S/(f) is a noncommutative central conic. By passing to the quadratic dual A! and locating a regular central element of degree 1 there, the authors define a dehomogenization D_z(A) that produces a noncommutative affine pencil R=S/(f₁,f₂). Conversely, homogenizing such a pencil yields back a central conic (Theorem 5.7). This establishes a bijection between the set A_z^{3,1} of graded algebras with a regular central element of degree 2 and the set A^{∨2,2} of affine pencils of conics.

The crucial bridge to Frobenius algebras comes from the construction C(A):=A!