Rational curves and ruled orders on surfaces

TL;DR

This paper investigates ruled orders on surfaces, describing their fibers as rational, analyzing the Hilbert scheme of rational curves, and providing evidence that these are examples of non-commutative ruled surfaces.

Contribution

It characterizes fibers of ruled orders, computes the Hilbert scheme of rational curves, and links ruled orders to non-commutative ruled surfaces, advancing understanding in non-commutative algebraic geometry.

Findings

Fibers of ruled orders are rational in a certain sense.

The Hilbert scheme of rational curves on ruled orders is determined.

Ruled orders serve as examples of non-commutative ruled surfaces.

Abstract

We 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.