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.