
# Atlas of $\Spec\mathbb{Z}[x]$

This is a collection of "charts" for the affine scheme $\Spec\mathbb{Z}[x]$. I have collected these out of a certain fascination for this object, and its various visualizations. For an in-depth discussion of this object and an important visualization I refer you to Lieven Le Bruyn's blog post titled Mumford's treasure map. If you have a map to contribute, or any related information, please contact me! The maps are ordered chronologically by publication date.

### 1966, David Mumford, Lectures on Curves on an Algebraic Surface

Surprisingly (at least to me) there is a map that predates the canonical map of $\Spec\mathbb{Z}[x]$ in print by at least one year (maybe three, as David Mumford himself gives 1964 as the publication date). Lieven Le Bruyn has written about this at Google+.

It is published in his Lectures on Curves on an Algebraic Surface, volume 59 of Annals of mathematics studies by Princeton Press. You can find it on page 28 of the book.

Remark that is actually a depiction of $\Proj\mathbb{Z}[x,y]$.

### 1967, David Mumford, Introduction to algebraic geometry: Preliminary version of first 3 Chapters

This is probably the most well-known map because it has been covered in Mumford's treasure map, and apparently some guy even made a LaTeX version of it. You can find it (if you have a precious copy of the original book) on page 141.

### 1988, David Mumford, The Red Book of Varieties and Schemes

I haven't been able to check this particular edition, which is the first reprint of David Mumford's classical book. It is named after the looks of the original 60s edition, which was a set of mimeographed notes bound together in a red cover. It is number 1358 in Springer's Lecture Notes in Mathematics, just like the 1999 reprint. The map is therefore listed under 1999, David Mumford, The Red Book of Varieties and Schemes. I presume they are completely similar.

### 1995, Miles Reid, Undergraduate Commutative Algebra

Finally, a map that hasn't been drawn by David Mumford! In his undergraduate algebra handbook he gives the following impression on page 24

You should check the frontispiece of this book by the way, which gives a really interesting viewpoint on the phrase "Let $A$ be a ring and $M$ and $A$-module..."

### 1999, David Mumford, The Red Book of Varieties and Schemes

In this reprint we find an updated version of the map of 1967, David Mumford, The Red Book of Varieties and Schemes. It is still number 1358 of Lectures Notes in Mathematics, and the map can be found on page 75.

### 2000, David Eisenbud and Joe Harris, The Geometry of Schemes

In their book The Geometry of Schemes, published by Springer in the series Graduate texts in mathematics as part 197 we find on page 85 their interpretation of a map of $\Spec\mathbb{Z}[x]$.

On page 86 we find an impression of the subscheme $\Spec\mathbb{Z}[x]/(x^2-3)$, which makes this page even more atlas-like.

### 2009, Lieven Le Bruyn, (non)commutative f-un geometry

In his preprint (non)commutative f-un geometry (arXiv:0909.2522) Lieven Le Bruyn develops some of the ideas from the post Manin's geometric axis (which serves as a follow-up on Mumford's treasure map). It contains the first full-colour impression, see page 8! Interesting fact about David Mumford, according to Wikipedia he is colour blind.

The original version of this map that focuses on the arithmetic nature of $\Spec\mathbb{Z}[x]$ is a modified version of Mumford's 1967 map and is discussed by Lieven Le Bruyn Manin's geometric axis.

### 2015, David Mumford and Tadao Oda, Algebraic Geometry II

The (penultimate) draft version of their book contains an updated version of Mumford's original drawing on page 120.

Thanks to Lieven Le Bruyn for his scans!