# 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 unpublished notes (1992) 53 pp.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.