Classification of noncommutative central conics
Haigang Hu, Izuru Mori, Wenchao Wu
TL;DR
The paper resolves the open problem of classifying noncommutative central conics by developing a comprehensive framework of noncommutative homogenization and dehomogenization. It proves bijections among isomorphism classes of $4$-dimensional Frobenius algebras, noncommutative affine pencils of conics, and noncommutative central conics, thereby reducing central-conic classifications to previously understood classifications of Frobenius algebras and pencils. Central to the approach is the introduction of $C(A)=A^![(f^!)^{-1}]_0$ and the use of duality, twists, and Bezout-type results to connect hypersurfaces, their duals, and their homogenizations. The results yield an explicit classification of noncommutative central conics, characterize geometric pairs, and provide practical criteria for identifying regular/central degree-1 elements in associated algebras, offering a robust toolkit for noncommutative quadric hypersurfaces with potential broader impact in noncommutative algebraic geometry.
Abstract
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.