  In this paper we consider an effective divisor on the complex projective line
and associate with it the module D consisting of all the derivations $\theta$
such that $\theta(I_i)\subset I_i^{m_i}$ for every $i$, where $I_i$ is the
ideal of $p_i$. The module D is graded and free of rank 2; the degrees of its
homogeneous basis, called the exponents, form an important invariant of the
divisor. We prove that under certain conditions on $(m_i)$ the exponents do not
depend on $\{p_i\}$. Our main result asserts that if these conditions do not
hold for $(m_i)$ then there exists a general position of $n$ points for which
the exponents do not change. We give an explicit formula for them. We also
exhibit some examples of degeneration of the exponents, in particular those
where the degeneration is defined by vanishing of certain Shur functions. As
application and motivation, we show that our results imply Terao's conjecture
(about the combinatorial nature of the freeness of hyperplane arrangements) for
certain new classes of arrangements of lines in the complex projective plane.
