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INVARIANT CURVES BY VECTOR FIELDS ON ALGEBRAIC VARIETIES

Published online by Cambridge University Press:  30 October 2000

A. CAMPILLO ,
M. M. CARNICER  and
J. GARCÍA DE LA FUENTE
A. CAMPILLO
Affiliation:
Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias, Universidad de Valladolid, Prado de la Magdalena s.n., 47005 Valladolid, Spain
M. M. CARNICER
Affiliation:
Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias, Universidad de Valladolid, Prado de la Magdalena s.n., 47005 Valladolid, Spain
J. GARCÍA DE LA FUENTE
Affiliation:
Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias, Universidad de Valladolid, Prado de la Magdalena s.n., 47005 Valladolid, Spain
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Abstract

If C is a reduced curve which is invariant by a one-dimensional foliation [Fscr ] of degree d[Fscr ] on the projective space then it is shown that d[Fscr ]−1+a is a bound for the quotient of the two coefficients of the Hilbert–Samuel polynomial for C, where a is an integer obtained from a concrete problem of imposing singularities to projective hypersurfaces, and so a bound is obtained for the degree of C when it is a complete intersection. Concrete values of a can be derived for several interesting applications. The results are presented in the form of intersection-theoretical inequalities for one-dimensional foliations on arbitrary smooth algebraic varieties.

Type
Research Article
Information
Journal of the London Mathematical Society , Volume 62 , Issue 1 , August 2000 , pp. 56 - 70
Copyright
The London Mathematical Society 2000

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