  We introduce a natural generalization of the scattering equations, which
connect the space of Mandelstam invariants to that of points on
${\mathbb{CP}^1}$, to higher-dimensional projective spaces $\mathbb{CP}^{k-1}$.
The standard, $k=2$ Mandelstam invariants, $s_{ab}$, are generalized to
completely symmetric tensors $\textsf{s}_{a_1a_2\ldots a_k}$ subject to a
`massless' condition $\textsf{s}_{a_1a_2\cdots a_{k-2}\,b\,b}=0$ and to
`momentum conservation'. The scattering equations are obtained by constructing
a potential function and computing its critical points. We mainly concentrate
on the $k=3$ case: study solutions and define the generalization of biadjoint
scalar amplitudes. We compute all `biadjoint amplitudes' for $(k,n)=(3,6)$ and
find a direct connection to the tropical Grassmannian. This leads to the notion
of $k=3$ Feynman diagrams. We also find a concrete realization of the new
kinematic spaces, which coincides with the spinor-helicity formalism for $k=2$,
and provides analytic solutions analogous to the MHV ones.
