Blowing up of non-commutative smooth surfaces (Q2755132)

In this book, the author deals with certain Abelian categories, which may be viewed as non commutative analogs of algebraic surfaces. For surfaces in that sense, a non commutative analog of the blow up at a point in the zero-divisor of a Poisson bracket on a surface is developed. NEWLINENEWLINENEWLINEThe book starts with a detailed introduction, which also explains relations to questions on graded algebras of Gelfand-Kirillov dimension three, that motivate many developments in the book. After a short review of some preliminaries on Abelian categories, the author develops a general categorical approach to non commutative analogs of several features of algebraic geometry, which forms the basis for the further developments. In particular, non commutative versions of quasi schemes and divisors are introduced and studied and the concept of the Rees algebra is developed in a general setting. NEWLINENEWLINENEWLINEStarting with a Noetherian quasi scheme \(X\) which contains a commutative curve \(Y\) as a divisor and satisfies an appropriate smoothness condition, the next two chapters study the formal neighborhood of a point \(p\in Y\). Some background from the theory of pseudo-compact rings is developed and the main results are based on an analysis of various completion functors. In the above setting, the author next introduces the blow up \(\widetilde X\) of \(X\) at a point \(p\in Y\), and studies its basic properties. It is shown that there is an analog of the exceptional curve and that \(\widetilde X\) and \(X\) are birational, thus supporting the analogy to the commutative case. After a short chapter containing some background on derived categories, the author next studies the derived category of \(\widetilde X\) which is shown to admit a canonical semi-orthogonal decomposition. NEWLINENEWLINENEWLINEThe remaining three chapters are motivated by the construction of Del Pezzo surfaces by blowing up collections of points in a projective plane in the commutative case. The quantum versions of projective planes considered by the author correspond to the regular algebras associated to certain elliptic triples in the commutative projective plane as introduced by \textit{M. Artin, J. Tate} and \textit{M. van den Bergh} [``Some algebras associated to automorphisms of elliptic curves'' in ``The Grothendieck Festschrift, Vol. I'', Prog. Math. 86, 33-85 (1990; Zbl 0744.14024)]. An analog of blowing up at up to eight points is established as a sequence of non commutative blow ups and derived equivalences. Results on the number of exceptional simple objects in the quasi schemes showing up in this sequence are presented. Finally, it is proved that the quasi scheme obtained by blowing up six points as described above is contained as a cubic divisor in a quantum version of projective three space.