Diptych varieties

Gavin Brown and Miles Reid

This website contains material on diptych varieties and their application to Mori flips of Type A. The paper [BR1] appears in the Proc. of the LMS. [BR2] will appear in Adv. Stud. in Pure Math. [BR3] and [BR4] are currently in preparation. See [BR1], Section 6 for their status in the project as a whole.

Our Magma files lr.m and diptych.m can help with the calculations. They come with instructions.

Our diptych papers

These are not stable versions. We update them from time to time. The date stamp below is a clue, but only a clue.

[BR1] Diptych varieties, I. Proc. London Math. Soc (107) (2013), 1353--1394. Available at arXiv:1208.2446

[BR2] Diptych varieties. II: Polar varieties. 32pp. To appear in Adv. Stud. in Pure Math., Kawamata's 60th volume. Available at arXiv:1208.5858

This handles the cases [2,2] and [4,1] as pullbacks from polar varieties, and also the small cases de < 4, some with polar interpretations.
The file polar.m contains Magma code that checks the calculations in Section 3.

[BR3] Diptych varieties. III: Redundant generators. 10pp.
This addresses the general case de > 4 when d = 1 or e = 1.

[BR4] Diptych varieties. IV: Mori flips of type A. 18pp.
Outlines the connection to flips and Mori's paper below

Homework exercises in diptych varieties.

Other papers

These papers are either used in our diptych papers or serve as motivation.

[Mo00] S. Mori, On semistable extremal neighborhoods, in Higher dimensional birational geometry (Kyoto, 1997), Adv. Stud. Pure Math. {\bf35}, Math. Soc. Japan, Tokyo, 2002, 157--184

[Re92] M. Reid, What is a flip? unpublished notes (1992) 53 pp.

There was a change on Mon 7 Sept 2015.