I'm trying to figure out what Harris wanted to say in following construction, called Chow construction (Algebraic Geometry by Joe Harris, p. 269):
The first construction of the parameter space for varieties of a given degree and dimension is called the Chow construction. The basic idea behind it is simply that the problem in parametrizing varieties $X \subset \mathbb{P}^n$ in general is that $X$ is not generally given by a single polynomial, whose coefficients we can vary freely as in the example of hypersurfaces. The point of the construction is thus to associate to any such variety $X$ a hypersurface $\Phi_X$, albeit not one in a projective space.
There are two essentially equivalent ways of doing this. Suppose first of all that $X$ has pure dimension $k$, and consider the incidence correspondence consisting of points $p \in X$ together with $(k + 1)$-tuples of hyperplanes containing $p$; that is,
$$ \Gamma = \{(p, H_1,..., H_{k+1}): \ p \in H_i \forall i \} $$ $$ \subset X \times (\mathbb{P}^{n})^* \times ... \times (\mathbb{P}^{n})^* $$
($(\mathbb{P}^{n})^*$ is the dual projective space) for each point $p \in X$, the set of hyperplanes containing $p$ is a hyperplane $\mathbb{P}^{n-1} \subset (\mathbb{P}^{n})^* $, so that the set of $(k + 1)$-tuples of hyperplanes is irreducible of dimension $(k + 1) \cdot (n - 1)$. We deduce that $\Gamma$ is of pure dimension $k + (k + 1) \cdot (n - 1) = (k + 1)· n - 1$, with one irreducible component corresponding to each irreducible component of $X$.
For a general choice of point $p \in X$ and $H_1,..., H_{k+1}$ containing $p$ the intersection of the $H_i$ with $X$ will consist only of the point $p$, so that the canonical projection map $\pi: \Gamma \to (\mathbb{P}^{n})^* \times ... \times (\mathbb{P}^{n})^* $ will be birational. It follows that the image of $\Gamma$ under this projection is a hypersurface in $(\mathbb{P}^{n})^* \times ... \times (\mathbb{P}^{n})^* $ (this is where we need X to have pure dimension); we will call this hypersurface $\Phi_X$.
Question: The last sentences I not understand. If a map between $f: X \to Y$ varieties/ schemes of pure dimension is birational then the image (and also it's closure in $Y$) cannot be contained in a subvariety $W \subset Y$ of smaller dimension than $\dim Y (= \dim \ X$ as $f$ birational). But a hypersurface of $Y$ has properly smaller dimension than $Y$ so the map cannot be birational.
Therefore I not understand why the projection map $\pi: \Gamma \to (\mathbb{P}^{n})^* \times ... \times (\mathbb{P}^{n})^* $ is birational and a hypersurface of $(\mathbb{P}^{n})^* \times ... \times (\mathbb{P}^{n})^* $ contains the image of $\Gamma$.