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Homethin algebraic set

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# 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 $W\neq V$, 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 $\phi\colon V^{{\prime}}\to V$ 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.

## Mathematics Subject Classification

12E25*no label found*

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