Consider an algebraically closed field $k$, and define $\mathbb P^n_k:=\textrm{Proj}(k[T_0,\ldots,T_n])$. In some algebraic geometry books I see the notation ${(\mathbb P^n_k)}^\vee$ that is referred as "the dual projective space" without any other precisation.
Now I'm confused: $\mathbb P^n_k$ is a scheme and I don't understand what is the formal meaning of its "dual". Maybe hyperplane divisors are involved?
Many thanks in advance.
If you have a $k$-vector space $V$ you can form the symmetric algebra on $V$, $$\operatorname{Sym} V = k \oplus V \oplus V^{\otimes 2} / S_2 \oplus V^{\otimes 3} / S_3 \oplus \cdots $$ and it is clearly a graded $k$-algebra. We define $\mathbb{P} (V) = \operatorname{Proj} \operatorname{Sym} V$. The dual of $\mathbb{P} (V)$ is just $\mathbb{P} (V^\vee)$.