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*,*X*^{2},*X*^{3}, …

satisfying certain axioms.

(N) Each of the *X*^{n} is a Noetherian topological space, of dimension at most *n*.

Some standard terminology for Noetherian spaces will now be assumed.

(A) In each *X*^{n}, the subsets defined by equality in an *n*-tuple are closed. The mappings

*X*^{m}→*X*^{n}

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

(B) For a projection

*p*:*X*^{m}→*X*^{n}

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*× *X*², with the following properties:

(I) Given pairs (*x*, *y*), (*x*′, *y*′) in *X*², 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 *X*², (*t*, *x*, *y*) in *Q* is an irreducible closed set of dimension 1.

(K) For all pairs (*x*, *y*), (*x*′, *y*′) in *X*², 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.

References

Hrushovski, Ehud; Zilber, Boris (1996). "Zariski Geometries" (PDF). Journal of the American Mathematical Society. 9 (01): 1–56. doi:10.1090/S0894-0347-96-00180-4.

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