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 space with the usual Zariski topology.

1. 1.

A subset $A\subset V(K)$ is said to be of type $C_{1}$ if there is a closed subset $W\subset V$, with $W\neq V$, such that $A\subset W(K)$. In other words, $A$ is not dense in $V$ (with respect to the Zariski topology).

2. 2.

A subset $A\subset V(K)$ is said to be of type $C_{2}$ if there is an irreducible variety $V^{\prime}$ of the same dimension as $V$, and a (generically) surjective algebraic morphism $\phi\colon V^{\prime}\to V$ of degree $\geq 2$, with $A\subset\phi(V^{\prime}(K))$

Example.

Let $K$ be a field and let $V(K)=\mathbb{A}(K)=\mathbb{A}^{1}(K)=K$ 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 $C_{1}$ are subsets formed by a finite number of points.

Let $V^{\prime}(K)=\mathbb{A}(K)$ be affine space and define:

 $\phi\colon V^{\prime}\to V$

by $\phi(k)=k^{2}$. Then $\deg(\phi)=2$. Thus, the subset:

 $A=\{k^{2}:k\in\mathbb{A}(K)\}$

, i.e. $A$ is the subset of perfect squares in $K$, is a subset of type $C_{2}$.

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 $C_{1}$ and type $C_{2}$.

References

• 1 J.-P. Serre, Topics in Galois Theory, Research Notes in Mathematics, Jones and Barlett Publishers, London.
Title thin algebraic set ThinAlgebraicSet 2013-03-22 15:14:13 2013-03-22 15:14:13 alozano (2414) alozano (2414) 5 alozano (2414) Definition msc 12E25 thin set mince set