Symmetric noncommutative birational transformations (Q1656217)

The classification of (commutative) projective surfaces is well-known. An analogous classification of ``noncommutative surfaces'' has been in progress for over 20 years, with goals and conjectures described in [\textit{M. Artin}, Lond. Math. Soc. Lect. Note Ser. 238, 1--19 (1997; Zbl 0888.16025)]. The paper under review makes important progress on ``birational transformations'' from noncommutative \(\mathbb{P}^1 \times \mathbb{P}^1\) to noncommutative \(\mathbb{P}^2\), showing that these transformations are invertible to an extent similar to the commutative case. More specifically, the paper studies Sklyanin algebras of dimension \(3\) over an algebraically closed field \(k\), a special case of Artin-Schelter regular algebras. Such an algebra is a coordinate ring \(A\) of a noncommutative \(\mathbb{P}^1 \times \mathbb{P}^1\) if the generators of \(A\) satisfy cubic relations or a coordinate ring \(A'\) of a noncommutative \(\mathbb{P}^2\) if the generators of \(A'\) satisfy quadratic relations. The category of graded algebras is not general enough, so the main results depend on \(\mathbb{Z}\)-algebras, ``roughly speaking, infinite matrix rings associated to graded rings'' [\textit{S. Sierra}, Algebr. Represent. Theory 14, No. 2, 377--390 (2011; Zbl 1258.16047)]. There are three main results. First, given a cubic Sklyanin \(\mathbb{Z}\)-algebra \(A\), there is a quadratic Sklyanin \(\mathbb{Z}\)-algebra \(A'\) and inclusion \(A' \hookrightarrow A^{(2)}\) inducing an isomorphism between their function fields. Given the contravariant correspondence between rings and schemes, this corresponds to a birational transformation \(\mathbb{P}^1 \times \mathbb{P}^1 \dashrightarrow \mathbb{P}^2\). This theorem extends results of [\textit{D. Presotto} and \textit{M. Van den Bergh}, J. Noncommut. Geom. 10, No. 1, 221--244 (2016; Zbl 1371.14005)] to the \(\mathbb{Z}\)-algebra case. The second main result is a kind of converse to the first, giving a noncommutative birational transformation \(\mathbb{P}^2 \dashrightarrow \mathbb{P}^1 \times \mathbb{P}^1\), which did not appear in the previous paper. The extension to \(\mathbb{Z}\)-algebras allows the inclusion of the noncommutative quadrics of [\textit{M. Van den Bergh}, Int. Math. Res. Not. 2011, No. 17, 3983--4026 (2011; Zbl 1311.14003)]. The third main result shows that any quadratic transform (described in Section 5), is in some sense invertible. The birational transformations of the previous paragraph are quadratic transforms, as is a noncommutative version of the Cremona transform. The paper is fairly self-contained, reviewing material on noncommutative projective schemes, birational transformations, Sklyanin algebras, \(\mathbb{Z}\)-algebras, \(\mathbb{Z}\)-domains, and \(\mathbb{Z}\)-field of fractions. A few proofs refer back to [\textit{D. Presotto} and \textit{M. Van den Bergh}, J. Noncommut. Geom. 10, No. 1, 221--244 (2016; Zbl 1371.14005)], but the great majority of proofs are original to this paper and completely spelled out.