Zariski geometry
Encyclopedia
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 surface
s in particular.
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
defined by projecting out certain coordinates and setting others as constants are all continuous.
(B) For a projection
and an irreducible closed subset Y of Xm, 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 Xm, other than the cases where the fiber is X.
(E) 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 (cf. very ample line bundle). 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:
(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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Zariski geometry consists of an abstract structure introduced by Ehud Hrushovski
Ehud Hrushovski
Ehud Hrushovski is a mathematical logician. He is a Professor of Mathematics at the Hebrew University of Jerusalem.His father, Benjamin Harshav, is Emeritus Professor in Yale University and Tel Aviv University to Comparative Literature and a poet....
and Boris Zilber, in order to give a characterisation of the Zariski topology
Zariski topology
In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
on an algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
, and all its powers. The Zariski topology on a product of algebraic varieties is very rarely the product topology
Product topology
In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called 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 surface
Compact Riemann surface
In mathematics, a compact Riemann surface is a complex manifold of dimension one that is a compact space. Riemann surfaces are generally classified first into the compact and the open .A compact Riemann surface C that is a...
s 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
Noetherian topological space
In mathematics, a Noetherian topological space is a topological space in which closed subsets satisfy the descending chain condition. Equivalently, we could say that the open subsets satisfy the ascending chain condition, since they are the complements of the closed subsets...
, 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
Tuple
In mathematics and computer science, a tuple is an ordered list of elements. In set theory, an n-tuple is a sequence of n elements, where n is a positive integer. There is also one 0-tuple, an empty sequence. An n-tuple is defined inductively using the construction of an ordered pair...
are closed. The mappings
- Xm → Xn
defined by projecting out certain coordinates and setting others as constants are all continuous.
(B) For a projection
- p: Xm → Xn
and an irreducible closed subset Y of Xm, p(Y) lies between its closure Z and Z \ Z′ where Z′ is a proper closed subset of Z. (This is quantifier elimination
Quantifier elimination
Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science. One way of classifying formulas is by the amount of quantification...
, 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 Xm, other than the cases where the fiber is X.
(E) 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 (cf. very ample line bundle). 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:
(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
Parametric family
In mathematics and its applications, a parametric family or a parameterized family is a family of objects whose definitions depend on a set of parameters....
.
Then Hrushovski and Zilber prove that under these conditions there is an algebraically closed field
Algebraically closed field
In mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F.-Examples:...
K, and a non-singular algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
C, such that its Zariski geometry of powers and their Zariski topology
Zariski topology
In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
is isomorphic to the given one. In short, the geometry can be algebraized.