The geometry of $\mathbb{F}_1$

Question

Over the years I have heard some people talking about a kind of geometry of $\mathbb{F}_1$. I will be very honest and admit that it does not make any sense to me.

The set $\mathbb{F}_1^n$ has precisely one point, and the variety associated to any polynomial is precisely the whole space. Everything looks trivial to me.

What are they really talking about?

Answer 1

You might be interested in the answers from a similar question on Math overflow: https://mathoverflow.net/questions/2300/what-is-the-field-with-one-element

The main things to note are that

1. there is no actual field with one element
2. there are many approaches to defining it
3. these approaches give various objects that "would be nice" to have in certain contexts, and generalise geometrical concepts in some appealing way