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. A subset $A\subset V(K)$ is said to be of type $C_1$ if there is a closed subset $W\subset V$, with $W\ne V$, such that $A\subset W(K)$. In other words, $A$ is not dense in $V$ (with respect to the Zariski topology).

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 $\varphi :V^{\prime }\to V$ of degree $\ge 2$, with $A\subset \varphi (V^{\prime }(K))$

Example.

Let $K$ be a field and let $V(K)=𝔸(K)={𝔸}^1(K)=K$ 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 $V^{\prime }(K)=𝔸(K)$ be affine space and define:

$$\varphi :V^{\prime }\to V$$

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

$$A=\{k^2:k\in 𝔸(K)\}$$

, 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 $V\mathrm{/}K$ 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 $C_{\mathrm{1}}$ and type $C_{\mathrm{2}}$.