Noncommutative versions of some classical birational transformations

TL;DR

This paper extends classical birational transformations to non-commutative algebraic geometry, demonstrating that certain non-commutative surfaces share the same function field and constructing a non-commutative Cremona transform.

Contribution

It introduces non-commutative analogues of classical birational transformations, including an analogue of the Cremona transform for non-commutative projective planes.

Findings

3D quadratic Sklyanin algebras share the same function field as 3D cubic Sklyanin algebras

Constructed a non-commutative Cremona transform

Established non-commutative versions of classical birational transformations

Abstract

In this paper we generalize some classical birational transformations to the non-commutative case. In particular we show that 3-dimensional quadratic Sklyanin algebras (non-commutative projective planes) and 3-dimensional cubic Sklyanin algebras (non-commutative quadrics) have the same function field. In the same vein we construct and analogue of the Cremona transform for non-commutative projective planes.