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 be a morphism of nonsingular varieties over an algebraically closed field. Under certain conditions, one can define a cycle on with .

The multiplicity of a component of can be computed directly from local equations for . If , and if is induced by projection from a suitable linear subspace of , then is , up to rational equivalence, where is the normal bundle of in .

(2) Let be a smooth projective morphism of noetherian schemes, where is connected, and the fibres of are absolutely irreducible -dimensional varieties. For a geometric point , and a locally free sheaf on , let be the corresponding geometric fibre, and the sheaf induced on . If are locally free sheaves on , and if , then the degree of the zero-cycle is independent of the choice of .

(3) The results of (1) and (2) are used to study the behavior under specialization of a closed subvariety which is the image under generic projection of a nonsingular .

**[1]***Séminaire C. Chevalley**année*: 1958.*Anneaux de Chow et applications*, Secrétariat mathématique, Paris, 1958. MR**22**#1572.**[2]**H. Fitting,*Die Determinantenideale eines Moduls*, Jber. Deutsch. Math.-Verein.**1936**, 195-228.**[3]**J. Fogarty and D. S. Rim,*Serre sequences and Chern classes*, J. Algebra**10**(1968), 436–447. MR**0241431****[4]**Alexander Grothendieck,*La théorie des classes de Chern*, Bull. Soc. Math. France**86**(1958), 137–154 (French). MR**0116023****[5]**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****[6]**Arthur Mattuck,*Secant bundles on symmetric products*, Amer. J. Math.**87**(1965), 779–797. MR**0199196****[7]**David Mumford,*Lectures on curves on an algebraic surface*, With a section by G. M. Bergman. Annals of Mathematics Studies, No. 59, Princeton University Press, Princeton, N.J., 1966. MR**0209285****[8]**Joel Roberts,*Generic projections of algebraic varieties*, Amer. J. Math.**93**(1971), 191–214. MR**0277530**

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1972-0306199-7

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