A class of quantum Zariski geometries

Abstract

This paper is an attempt to understand the nature of non-classical Zariski geometries. Examples of such structures were first discovered in [HZ]. These examples showed that contrary to some expectations, one-dimensional Zariski geometries are not necessarily algebraic curves. Given a smooth algebraic curve C with a big enough group of regular automorphisms, one can produce a “smooth” Zariski curve C along with a finite cover p : C → C. C cannot be identified with any algebraic curve because the construction produces an unusual subgroup of the group of regular automorphisms of C ([HZ], section 10). The main theorem of [HZ] states that every Zariski curve has the form C, for some algebraic C. So, only in the limit case, when p is bijective, is the curve algebraic. A typical example of an unusual subgroup of the automorphism group of such a C is the nilpotent group of two generators U and V with the central commutator 2 = [U,V] of finite order N. So, the defining relations are