# Contents

## Idea

Geometric stability theory is the principal part of what is called geometric model theory?. It was introduced in works of Boris Zilber, Cherlin, Ehud Hrushovski, and Anand Pillay.

## Basic concepts

###### Definition

Let $X$ be a set. A pregeometry on $X$ is a closure operator (i.e., a monad $\mathrm{cl}:PX\to PX$ on the power set), satisfying the following two conditions:

• The monad $\mathrm{cl}$ is finitary, i.e., $A\in X$ and $a\in \mathrm{cl}\left(A\right)$, then there is a finite ${A}_{0}\subseteq A$ such that $a\in \mathrm{cl}\left({A}_{0}\right)$.

• (Exchange condition) If $A\in PX$, $a,b\in X$, and $a\in \mathrm{cl}\left(A\cup \left\{b\right\}\right)$, then $a\in \mathrm{cl}\left(A\right)$ or $b\in \mathrm{cl}\left(A\cup \left\{a\right\}\right)$. (Cf. matroid)

A geometry is a pregeometry such that $\mathrm{cl}\left(\varnothing \right)=\varnothing$ and $\mathrm{cl}\left(\left\{x\right\}\right)=\left\{x\right\}$ for all $x\in X$.

###### Examples
• Let $X$ be a vector space, and let $\mathrm{cl}$ be the monad on $PX$ whose algebras are vector subspaces of $X$. Clearly $\mathrm{cl}$ is finitary (any subspace is the set-theoretic union of finite-dimensional subspaces), and the exchange condition is a classical fact about vector spaces related to the notion of independence. Thus $\mathrm{cl}$ is a pregeometry.

• Similarly, let $X$ be a projective space $ℙV$, and let $\mathrm{cl}$ be the monad on $PX$ whose algebras are projective subspaces. Then $\mathrm{cl}$ is a geometry (the closure of a point is a point). Any pregeometry $\mathrm{cl}$ gives rise to a geometry in a similar way, in the sense that a pregeometry $\mathrm{cl}$ induces a geometry on the image of the function $X\to PX$, $x↦\mathrm{cl}\left(\left\{x\right\}\right)$.

• Let $X$ be an algebraically closed field; let $\mathrm{cl}$ be the monad on $PX$ whose algebras are algebraically closed subfields. Then $\mathrm{cl}$ is a pregeometry. That the exchange condition is satisfied is a result due to Steinitz.

###### Definition

Given a pregeometry $\left(X,\mathrm{cl}\right)$, a subset $A\in PX$ is independent if for all $a\in A$, $a\notin \mathrm{cl}\left(A-\left\{a\right\}\right)$. An independent set $A$ said to be a basis for $Y\in PX$ if $Y\subseteq \mathrm{cl}\left(A\right)$. All bases of $Y$ have the same cardinality (?), called the dimension of $Y$.

## References

• Anand Pillay, Geometric stability theory, Oxford Logic Guides 32
• slides from conference ”Geometric model theory”, Oxford 2010: directory html
• Misha Gavrilovich, Model theory of universal covering space of complex algebraic varieties, thesis, pdf
Revised on September 21, 2012 00:08:59 by Urs Schreiber (82.169.65.155)