The variation of singular cycles in an algebraic family of morphisms

Abstract:

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