# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## The variation of singular cycles in an algebraic family of morphismsHTML articles powered by AMS MathViewer

by Joel Roberts
Trans. Amer. Math. Soc. 168 (1972), 153-164 Request permission

## 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 ${V^r} \subset {P^n}$.
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