Invariant curves by vector fields on algebraic varieties

IF C is a reduced curve which is invariant by a one-dimensional foliation F of degree d(F) on the projective space then it is shown that d(F) - 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.