of $X$ with the hyperplanes defined by $\big\{ l_i = 0 \big\}$ for $i \in \{ 1, \ldots, n \}$ is called the *complex configuration set* of $\Pi$ and denoted by $K_\Pi$; the real points of this intersection are called the *real configuration set* of $\Pi$. The complex dimension of $K_\Pi \cap {\mathrm{SE}_{3,{\mathbb{C}}}}$ as a complex algebraic variety is called the *mobility* of $\Pi$. If the mobility of $\Pi$ is greater than or equal to $1$, then $\Pi$ is said to be *mobile*.

\[rem:cardinality\] For a generic hexapod, the complex configuration set is finite of cardinality $40$. This has been shown by [@RongaVust]. It also follows from the fact $\deg(X) = 40$ that was mentioned in Subsection \[compactification:new\]: in fact in the coordinates of ${\mathbb{P}}^{16}_{{\mathbb{C}}}$ the spherical equation becomes linear, hence every leg of an $n$–pod imposes a linear condition on $X$. For a generic hexapod $\Pi$, its complex configuration set $K_{\Pi}$ is given by the intersection of $6$ generic hyperplanes in ${\mathbb{P}}^{16}_{{\mathbb{C}}}$ with $X$. The intersection of the hyperplanes gives a generic codimension $6$ linear space $H_{\Pi}$. Now we use the following general fact from projective geometry: the intersection of a complex projective variety of dimension $r$ and degree 