## The variation of singular cycles in an algebraic family of morphisms

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- by Joel Roberts
- Trans. Amer. Math. Soc.
**168**(1972), 153-164 - DOI: https://doi.org/10.1090/S0002-9947-1972-0306199-7
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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 $\operatorname {Supp} ({S_i}) = \{ x|{\dim _{k(x)}}(\Omega _{X/Y}^1)(x) \geqq i\}$. 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 ${V^r} \subset {P^n}$.## References

- J. Fogarty and D. S. Rim,
*Serre sequences and Chern classes*, J. Algebra**10**(1968), 436–447. MR**241431**, DOI 10.1016/0021-8693(68)90071-9 - Alexander Grothendieck,
*La théorie des classes de Chern*, Bull. Soc. Math. France**86**(1958), 137–154 (French). MR**116023**, DOI 10.24033/bsmf.1501 - W. V. D. Hodge and D. Pedoe,
*Methods of algebraic geometry. Vol. II. Book III: General theory of algebraic varieties in projective space. Book IV: Quadrics and Grassmann varieties*, Cambridge, at the University Press, 1952. MR**0048065** - Arthur Mattuck,
*Secant bundles on symmetric products*, Amer. J. Math.**87**(1965), 779–797. MR**199196**, DOI 10.2307/2373245 - David Mumford,
*Lectures on curves on an algebraic surface*, Annals of Mathematics Studies, No. 59, Princeton University Press, Princeton, N.J., 1966. With a section by G. M. Bergman. MR**0209285**, DOI 10.1515/9781400882069 - Joel Roberts,
*Generic projections of algebraic varieties*, Amer. J. Math.**93**(1971), 191–214. MR**277530**, DOI 10.2307/2373457

*Séminaire C. Chevalley*$2{\text {e}}$

*année*: 1958.

*Anneaux de Chow et applications*, Secrétariat mathématique, Paris, 1958. MR

**22**#1572. H. Fitting,

*Die Determinantenideale eines Moduls*, Jber. Deutsch. Math.-Verein.

**1936**, 195-228.

## Bibliographic Information

- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**168**(1972), 153-164 - MSC: Primary 14E15
- DOI: https://doi.org/10.1090/S0002-9947-1972-0306199-7
- MathSciNet review: 0306199