Zariski geometry

Last updated November 17, 2022

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 sets

X, X², X³, ...

satisfying certain axioms.

(N) Each of the Xⁿ is a Noetherian topological space, of dimension at most n.

Some standard terminology for Noetherian spaces will now be assumed.

(A) In each Xⁿ, the subsets defined by equality in an n-tuple are closed. The mappings

X^m → Xⁿ

defined by projecting out certain coordinates and setting others as constants are all continuous.

(B) For a projection

p: X^m → Xⁿ

and an irreducible closed subset Y of X^m, p(Y) lies between its closure Z and Z \ Z′ where Z′ is a proper closed subset of Z. (This is quantifier elimination, at an abstract level.)

(C) X is irreducible.

(D) There is a uniform bound on the number of elements of a fiber in a projection of any closed set in X^m, other than the cases where the fiber is X.

(E) A closed irreducible subset of X^m, 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 (cf. very ample line bundle). It is assumed there is an irreducible closed subset P of some X^m, and an irreducible closed subset Q of P× X², with the following properties:

(I) Given pairs (x, y), (x′, y′) in X², for some t in P, the set of (t, u, v) in Q includes (t, x, y) but not (t, x′, y′)

(J) For t outside a proper closed subset of P, the set of (x, y) in X², (t, x, y) in Q is an irreducible closed set of dimension 1.

(K) For all pairs (x, y), (x′, y′) in X², selected from outside a proper closed subset, there is some t in P such that the set of (t, u, v) in Q includes (t, x, y) and (t, x′, y′).

Geometrically this says there are enough curves to separate points (I), and to connect points (K); 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.

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References

Hrushovski, Ehud; Zilber, Boris (1996). "Zariski Geometries" (PDF). Journal of the American Mathematical Society . 9 (1): 1–56. doi: 10.1090/S0894-0347-96-00180-4 .

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