Let $K$ be a field, and let $V$ be a finite-dimensional [vector space](/page/Vector%20Space) over $K$ with $\dim_K V \geq 2$. Let $V^* = \operatorname{Hom}_K(V,K)$ be the dual vector space. The map
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\begin{align*}
\Phi: \mathbb{P}(V^*) \to \{\mathbb{P}(W) : W \leq V \text{ and } \dim_K W = \dim_K V - 1\}
\end{align*}