Hilbert schemes for quantum planes are projective

非常一般的非交换射影平面上幻影范畴的不存在性

Nonexistence of phantom categories on very general noncommutative projective planes

本文证明了非常一般的非交换射影平面不容许幻影范畴。

Twisted rings and moduli stacks of "fat" point modules in non-commutative projective geometry

The Hilbert scheme of point modules was introduced by Artin-Tate-Van den Bergh to study non-commutative graded algebras. The key tool is the construction of a map from the algebra to a twisted ring on this Hilbert scheme. In this paper, we study moduli stacks of more general "fat" point modules, and show that there is a similar map to a twisted ring associated to the stack. This is used to provide a sufficient criterion for a non-commutative projective surface to be birationally PI. It is hoped that such a criterion will be useful in understanding Mike Artin's conjecture on the birational clas…

Sklyanin algebras and Hilbert schemes of points

We construct projective moduli spaces for torsion-free sheaves on noncommutative projective planes. These moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual (commutative) projective plane P^2. The generic noncommutative plane corresponds to the Sklyanin algebra S constructed from an automorphism sigma of infinite order on an elliptic curve E < P^2. In this case, the fine moduli space of line bundles over S with first Chern class zero and Euler characteristic (1-n) provides a sy…

点态非交换希尔伯特概形

Punctual noncommutative Hilbert schemes

该研究聚焦点态非交换希尔伯特概形，通过构造仿射铺筑与奇点小消解，确定了这类射影簇的 motive 与相交上同调。

Deformation of the Okubic Albert algebra and its relation to the Okubic affine and projective planes

We present a deformation of the Okubic Albert algebra introduced by Elduque whose rank-1 idempotent elements are in biunivocal correspondence with points of the Okubic projective plane, thus extending to Okubic algebras the correspondence found by Jordan, Von Neumann, Wigner between the octonionic projective plane and the rank-3 Exceptional Jordan Algebra.

Veronese representation of projective Hjelmslev planes over some quadratic alternative algebras

We geometrically characterise the Veronese representations of ring projective planes over algebras which are analogues of the dual numbers, giving rise to projective Hjelmslev planes of level 2 coordinatised over quadratic alternative algebras. These planes are related to affine buildings of relative type $\widetilde{A}_2$ and respective absolute type $\widetilde{\mathsf{A}}_2$, $\widetilde{\mathsf{A}}_5$ and $\widetilde{\mathsf{E}}_6$.