Line Bundles and Curves on a del Pezzo Order

TL;DR

This paper explicitly computes the Picard and Hilbert schemes of a specific noncommutative surface (a del Pezzo order) on the projective plane, revealing their geometric structure and relationship to genus two curves.

Contribution

It provides the first explicit computation of moduli spaces for a del Pezzo order on the projective plane, linking the Hilbert scheme to a genus two curve.

Findings

Hilbert scheme is a ruled surface over a genus two curve

Genus two curve is the Picard scheme of the order

Explicit examples of moduli spaces for noncommutative surfaces

Abstract

Orders on surfaces provide a rich source of examples of noncommutative surfaces. Other than some existence results, very little is known about the various moduli spaces that can be associated to them. Even fewer examples have been explicitly computed. In this paper we compute the Picard and Hilbert schemes of an order on the projective plane ramified on a union of two conics. Our main result is that, upon carefully selecting the right Chern classes, the Hilbert scheme is a ruled surface over a genus two curve. Furthermore, this genus two curve is, in itself, the Picard scheme of the order.