# New paper: Central curves on noncommutative surfaces

This blogpost is about
**Central curves on noncommutative surfaces**.
If you just want mathematics and not some of the backstory,
you can click the link to immediately get to the preprint.

### Background

Almost to the day 4 years ago, I wrote a blogpost about how to (possibly) interpret a type of noncommutative curve using the geometry of orders on varieties. I did so, because I was initially confused by some results in the literature, and by using the very different perspective outlined in that blogpost, I could make sense of the results in a way that made me really happy.

In a joint work with
my PhD student Thilo Baumann
and
former postdoc Okke van Garderen
we have now fully fleshed out the suggestion I made there,
and much more,
in our paper **Central curves on noncommutative surfaces**.

The context for our work is the dictionary between the geometry of orders on varieties and the geometry of (certain) Deligne–Mumford stacks. In the curve case, this is a result due to Chan–Ingalls from 2004. In the surface case, it is a recent result due to Faber–Ingalls–Okawa–Satriano, accepted for publication this year (but maybe only published in 2025?).

I am really proud of this work,
as I keep being fascinated about the *geometry*
in noncommutative algebraic geometry,
and I believe that this paper combines some interesting geometry
with some interesting noncommutative algebra.

### Overview of the results

The set of results in our paper concerns restrictions of orders to curves,
saying that one can restrict the equivalence of categories
given by the dictionary relating orders on surfaces to Azumaya algebras on certain Deligne–Mumford surfaces
to the *central curves* of the title:
curves on the underlying (stacky) surface.

Because the restriction can be *singular*,
even if the curve itself is smooth,
because of how the curve can interact with the ramification divisor of the order,
the resulting stack can be singular too,
and thus we extend the 1-dimensional Chan–Ingalls
beyond the hereditary case.

The second set of results in our paper revisit

- the noncommutative conics considered by Hu–Matsuno–Mori
- the skew Fermat curves considered by Kanazawa

by using the central Proj construction for a graded algebra finite over its center to obtain an order on a surface, and study these noncommutative curves using geometric methods. This is the goal that I originally outlined 4 years ago in my blogpost (when I only knew of Ueyama's special case of skew conics) but it turns out that our tools go greatly beyond this.

We can only flesh out the dictionary between singular orders on curves and singular stacks in explicit examples, and we would love to understand the complete picture. Let us know if you have any ideas!