The origins of this book lie in an extended visit that I made in the Spring of 2008 to MSRI in Berkeley, California as the Simons Professor for the programme on Representation Theory of Finite Groups and Related Topics. Jon Carlson and Julia Pevtsova were there for a large part of this time, and Eric Friedlander passed through town several times. Through numerous conversations and lectures, they ignited my interest in the theory of modules of constant Jordan type. At first, I was reluctant to be drawn in. But then I managed to prove one of the conjectures from a paper of Carlson, Friedlander and Pevtsova [81], and I was hooked. This work has been published [42] and appears here as Section 5.12.

After that initial success, I started working with Julia Pevtsova on the vector bundles on projective space associated to modules of constant Jordan type. We refined a previous definition of Friedlander and Pevtsova, and proved a realisation theorem. This work has also been published [53] and appears here as Theorem 8.1.1, Sections 8.2–8.9 and Section 10.2.

I wrote several other papers soon after that, and soon it was getting to the point where I had so much material that it made sense to make a book based on the resulting series of papers as well as a great deal of other unpublished work. This is that book.

My thanks go to MSRI for its hospitality in 2008 and 2013 while parts of this work were being written; David Eisenbud for sharing his knowledge of vector bundles on projective spaces; Jon Carlson, Eric Friedlander and Julia Pevtsova for numerous conversations; Serge Bouc, Radu Stancu and Jon Carlson for their extensive feedback on earlier versions of the text; Jeremy Rickard for various interesting comments and questions, and particularly for formulating Conjecture 5.13.1; Andrew Granville for helping me with some number theoretic questions, and especially for formulating and supplying a proof of a statement similar to Lemma 12.10.1. I'd also like to thank Jesse Burke, Jon Carlson, Claudia Miller, Julia Pevtsova and Greg Stevenson for enlightening conversations and feedback related to Chapter 11, and Mark Walker for giving a talk at a conference in Seattle that stimulated my interest in Orlov's correspondence.