Zariski geometry
In mathematics, a Zariski geometry consists of an abstract structure introduced by Ehud Hrushovski and Boris Zilber, in order to give a characterisation of the Zariski topology on an algebraic curve, and all its powers. The Zariski topology on a product of algebraic varieties is very rarely the product topology, but richer in closed sets defined by equations that mix two sets of variables. The result described gives that a very definite meaning, applying to projective curves and compact Riemann surfaces in particular.
Definition
A Zariski geometry consists of a set X and a topological structure on each of the setssatisfying certain axioms.
Each of the Xn is a Noetherian topological space, of dimension at most n.
Some standard terminology for Noetherian spaces will now be assumed.
In each Xn, the subsets defined by equality in an n-tuple are closed. The mappings
defined by projecting out certain coordinates and setting others as constants are all continuous.
For a projection
and an irreducible closed subset Y of Xm, p lies between its closure Z and Z \ ' where ' is a proper closed subset of Z.
X is irreducible.
There is a uniform bound on the number of elements of a fiber in a projection of any closed set in Xm, other than the cases where the fiber is X.
A closed irreducible subset of Xm, of dimension r, when intersected with a diagonal subset in which s coordinates are set equal, has all components of dimension at least r − s + 1.
The further condition required is called very ample. It is assumed there is an irreducible closed subset P of some Xm, and an irreducible closed subset Q of P× X2, with the following properties:
Given pairs, in X2, for some t in P, the set of in Q includes but not
For t outside a proper closed subset of P, the set of in X2, in Q is an irreducible closed set of dimension 1.
For all pairs , in X2, selected from outside a proper closed subset, there is some t in P such that the set of in Q includes and.
Geometrically this says there are enough curves to separate points, and to connect points ; and that such curves can be taken from a single parametric family.
Then Hrushovski and Zilber prove that under these conditions there is an algebraically closed field K, and a non-singular algebraic curve C, such that its Zariski geometry of powers and their Zariski topology is isomorphic to the given one. In short, the geometry can be algebraized.