Derivations of an effective divisor on the complex projective line

Abstract:

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 the vanishing of certain Schur functions. As an application and motivation, we show that our results imply Terao’s conjecture (concerning the combinatorial nature of the freeness of hyperplane arrangements) for certain new classes of arrangements of lines in the complex projective plane.