On the definition of noncommutative del Pezzo surfaces

Noncommutative projective planes and noncommutative quadrics have clear definitions as the category qgr of 3-dimensional Artin-Schelter regular quadratic (resp. cubic) Z-algebras. Other noncommutative del Pezzo surfaces lack a definition of this sort, but are constructed by blowing up noncommutative projective planes. In this talk I will talk about an attempt toward a definition of noncommutative del Pezzo surfaces without using blowups. This is a joint work in progress with Tarig Abdelgadir and Kazushi Ueda.