  We present several classes of constraints on the discontinuities of Feynman
integrals that go beyond the Steinmann relations. These constraints follow from
a geometric formulation of the Landau equations that was advocated by Pham, in
which the singularities of Feynman integrals correspond to critical points of
maps between on-shell spaces. To establish our results, we review elements of
Picard-Lefschetz theory, which connect the homotopy properties of the space of
complexified external momenta to the homology of the combined space of on-shell
internal and external momenta. An important concept that emerges from this
analysis is the question of whether or not a pair of Landau singularities is
compatible-namely, whether or not the Landau equations for the two
singularities can be satisfied simultaneously. Under conditions we describe,
sequential discontinuities with respect to non-compatible Landau singularities
must vanish. Although we only rigorously prove results for Feynman integrals
with generic masses in this paper, we expect the geometric and algebraic
insights that we gain will also assist in the analysis of more general Feynman
integrals.
