# $W=\mathbb{P}(S)$ for some (k+1)-dimensional subspace $S$ of $\mathbb{k}^{n+1}$

Let $$W$$ be a subset of $$\mathbb{P}_n(\mathbb{k})$$. It is known that for every affine map $$A$$ (with which $$W$$ intersects) $$A \cap W$$ is a k-dimensional affine space. Is it true that $$W=\mathbb{P}(S)$$ for some (k+1)-dimensional subspace $$S$$ of $$\mathbb{k}^{n+1}$$?