# Mori program for Brauer log pairs in dimension three

July 14 to July 18, 2014

at the

American Institute of Mathematics, Palo Alto, California

organized by

Daniel Chan, Kenneth Chan, Colin Ingalls, and Sandor Kovacs

This workshop, sponsored by AIM and the NSF, will be devoted to the Mori program for Brauer log pairs in dimension three.

The recent proof of the finite generation of the canonical ring is a major milestone in the Mori program for classifying higher dimensional varieties. The framework of the Mori program has been established in many different contexts, such as for log varieties, varieties with group actions, and rather surprisingly, for Brauer log pairs. These pairs arise naturally in the study of maximal orders and generic projective space bundles. The Mori program in the context of Brauer log pairs, which we call Brauer log MMP, has been completed in dimension two. The main focus for this workshop is to extend this work to dimension three and explore some applications.

The main topics for the workshop are

• Existence of terminal resolutions.
The notion of discrepancy generalises for Brauer log pairs, so we can define Brauer terminal pairs. Does every Brauer log pair admit a terminal resolution?

• Mori contractions.
What properties of a Brauer terminal pair are preserved under Mori contractions?
What is the right definition of a flip for Brauer log pairs?

• Classification of terminal pairs and étale local models.
Classify the ramification data associated to Brauer log pairs which are Brauer terminal.
Given a maximal order on a threefold germ $X$, we can associate a Brauer log pair. If we fix a Brauer log pair $(X,\alpha)$ which is Brauer terminal, can we classify the maximal orders such that the associated Brauer log pair is $(X,\alpha)$?

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.