  It is well-known that a Riemann surface can be decomposed into the so-called
pairs-of-pants. Each pair-of-pants is diffeomorphic to a Riemann sphere minus 3
points. We show that a smooth complex projective hypersurface of arbitrary
dimension admits a similar decomposition. The n-dimensional pair-of-pants is
diffeomorphic to the complex projective n-space minus n+2 hyperplanes.
  Alternatively, these decompositions can be treated as certain fibrations on
the hypersurfaces. We show that there exists a singular fibration on the
hypersurface with an n-dimensional polyhedral complex as its base and a real
n-torus as its fiber. The base accomodates the geometric genus of a
hypersurface V. Its homotopy type is a wedge of h^{n,0}(V) spheres S^n.
