Seminar

Noncommutative projective planes and noncommutative P1 \times P1 are given by certain classes of noncommutative algebras called Artin-Schelter regular 3-dimensional quadratic (resp. cubic) Z-algebras.  Moreover, one-to-one correspondences between these Z-algebras and certain commutative geometric data have been established by Artin, Schelter, Tate, Van den Bergh, Bondal, Polishchuk.  I will report on a joint project, largely in progress, which introduces broader classes of such algebras from which all noncommutative del Pezzo surfaces are obtained in a completely analogous way. If time permits I will try to describe what we have understood so far towards the goal of the project, which is to generalize the one-to-one correspondence as above to such algebras. This talk will be partly based on arXiv:2007.07620 and arXiv:2404.00175.