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Transactions of the American Mathematical Society

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



The variation of singular cycles in an algebraic family of morphisms

Author: Joel Roberts
Journal: Trans. Amer. Math. Soc. 168 (1972), 153-164
MSC: Primary 14E15
MathSciNet review: 0306199
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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\vert{\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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Keywords: Projective algebraic variety, generic projection, pinch point, rational equivalence ring, Chern class, locally free sheaf, Fitting ideal, singular cycle, noetherian scheme, smooth projective morphism, geometric fibre, specialization
Article copyright: © Copyright 1972 American Mathematical Society