Kenneth Chan
Rational curves and ruled orders on surfaces.
Fri, 07/01/2011 - 15:01 — SomNambulAuthors: Kenneth Chan, Daniel Chan
Subjects: Rings and Algebras
AbstractWe study ruled orders. These arise naturally in the Mori program for orders
on projective surfaces and morally speaking are orders on a ruled surface
ramified on a bisection and possibly some fibres. We describe fibres of a ruled
order and show they are in some sense rational. We also determine the Hilbert
scheme of rational curves and hence the corresponding non-commutative Mori
contraction. This gives strong evidence that ruled orders are examples of the
non-commutative ruled surfaces introduced by Van den Bergh.Log terminal orders are numerically rational.
Tue, 12/01/2009 - 16:01 — SomNambulAuthors: Kenneth Chan
Subjects: Algebraic Geometry
AbstractNoncommutative surfaces finite over their centres can be realised as orders
over surfaces. The aim of this paper is to present a noncommutative
generalisation of rational singularities, which we call numerical rationality,
for such orders. We show that numerical rationality is independent of the
choice of resolution. Our main result is that the log terminal orders arising
from the noncommutative minimal model program, in particular, canonical orders
are numerically rational. Both of these generalise well known facts about
rational singularities in commutative algebraic geometry.
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