(1) Let g : V r → W m ( m ≧ r ) g:{V^r} \to {W^m}(m \geqq r) be a morphism of nonsingular varieties over an algebraically closed field. Under certain conditions, one can define a cycle S i {S_i} on V V with Supp ( S i ) = { x | dim k ( x ) ( Ω X / Y 1 ) ( x ) ≧ i } \operatorname {Supp} ({S_i}) = \{ x|{\dim _{k(x)}}(\Omega _{X/Y}^1)(x) \geqq i\} . The multiplicity of a component of S i {S_i} can be computed directly from local equations for g g . If V r ⊂ P n {V^r} \subset {P^n} , and if g : V → P m g:V \to {P^m} is induced by projection from a suitable linear subspace of P n {P^n} , then S 1 {S_1} is c m − r + 1 ( N ⊗ O ( − 1 ) ) {c_{m - r + 1}}(N \otimes \mathcal {O}( - 1)) , up to rational equivalence, where N N is the normal bundle of V V in P n {P^n} . (2) Let f : X → S f:X \to S be a smooth projective morphism of noetherian schemes, where S S is connected, and the fibres of f f are absolutely irreducible r r -dimensional varieties. For a geometric point η : Spec ( k ) → S \eta :\operatorname {Spec} (k) \to S , and a locally free sheaf E E on X X , let X η {X_\eta } be the corresponding geometric fibre, and E η {E_\eta } the sheaf induced on X η {X_\eta } . If E 1 , … , E m {E_1}, \ldots ,{E_m} are locally free sheaves on X X , and if i 1 + ⋯ + i m = r {i_1} + \cdots + {i_m} = r , then the degree of the zero-cycle c i 1 ( E 1 η ) ⋯ c i m ( E m η ) {c_{{i_1}}}({E_{1\eta }}) \cdots {c_{{i_m}}}({E_{m\eta }}) is independent of the choice of η \eta . (3) The results of (1) and (2) are used to study the behavior under specialization of a closed subvariety V ′ ⊂ P 2 r − 1 V’ \subset {P^{2r - 1}} which is the image under generic projection of a nonsingular V r ⊂ P n {V^r} \subset {P^n} .