In this paper, we classify quadratic algebras in two variables at two levels: (1) up to isomorphism of graded algebras, (2) up to graded Morita equivalence. In general, it is difficult to classify algebras by looking at generators and relations, so we take a geometric approach, namely, using point schemes defined by Artin, Tate and Van den Bergh, to complete the classification.
"Geometric classification of quadratic algebras in two variables." Tsukuba J. Math. 34 (1) 79 - 96, August 2010. https://doi.org/10.21099/tkbjm/1283967409