thin algebraic set

Definition 1.

Let V be an irreducible algebraic variety (we assume it to be integral and quasi-projective) over a field K with characteristic zero. We regard V as a topological spaceMathworldPlanetmath with the usual Zariski topologyMathworldPlanetmath.

  1. 1.

    A subset AV(K) is said to be of type C1 if there is a closed subset WV, with WV, such that AW(K). In other words, A is not dense in V (with respect to the Zariski topology).

  2. 2.

    A subset AV(K) is said to be of type C2 if there is an irreducible variety V of the same dimensionMathworldPlanetmath as V, and a (generically) surjectivePlanetmathPlanetmath algebraic morphism ϕ:VV of degree 2, with Aϕ(V(K))


Let K be a field and let V(K)=𝔸(K)=𝔸1(K)=K be the 1-dimensional affine space. Then, the only Zariski-closed subsets of V are finite subsets of points. Thus, the only subsets of type C1 are subsets formed by a finite number of points.

Let V(K)=𝔸(K) be affine space and define:


by ϕ(k)=k2. Then deg(ϕ)=2. Thus, the subset:


, i.e. A is the subset of perfect squaresMathworldPlanetmath in K, is a subset of type C2.

Definition 2.

A subset A of an irreducible variety V/K is said to be a thin algebraic set (or thin set, or “mince” set) if it is a union of a finite number of subsets of type C1 and type C2.


  • 1 J.-P. Serre, Topics in Galois TheoryMathworldPlanetmath, Research Notes in Mathematics, Jones and Barlett Publishers, London.
Title thin algebraic set
Canonical name ThinAlgebraicSet
Date of creation 2013-03-22 15:14:13
Last modified on 2013-03-22 15:14:13
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 5
Author alozano (2414)
Entry type Definition
Classification msc 12E25
Synonym thin set
Synonym mince set