  We present a detailed description of the recent idea for a direct
decomposition of Feynman integrals onto a basis of master integrals by
projections, as well as a direct derivation of the differential equations
satisfied by the master integrals, employing multivariate intersection numbers.
We discuss a recursive algorithm for the computation of multivariate
intersection numbers and provide three different approaches for a direct
decomposition of Feynman integrals, which we dub the straight decomposition,
the bottom-up decomposition, and the top-down decomposition. These algorithms
exploit the unitarity structure of Feynman integrals by computing intersection
numbers supported on cuts, in various orders, thus showing the synthesis of the
intersection-theory concepts with unitarity-based methods and integrand
decomposition. We perform explicit computations to exemplify all of these
approaches applied to Feynman integrals, paving a way towards potential
applications to generic multi-loop integrals.
