  We prove an identity of Segre classes for zero-schemes of compatible sections
of two vector bundles. Applications include bounds on the number of equations
needed to cut out a scheme with the same Segre class as a given subscheme of
(for example) a projective variety, and a `Segre-Bertini' theorem controlling
the behavior of Segre classes of singularity subschemes of hypersurfaces under
general hyperplane sections.
  These results interpolate between an observation of Samuel concerning
multiplicities along components of a subscheme and facts concerning the
integral closure of corresponding ideals. The Segre-Bertini theorem has
applications to characteristic classes of singular varieties. The main results
are motivated by the problem of computing Segre classes explicitly and
applications of Segre classes to enumerative geometry.
