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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Derivations of an effective divisor on the complex projective line


Authors: Max Wakefield and Sergey Yuzvinsky
Journal: Trans. Amer. Math. Soc. 359 (2007), 4389-4403
MSC (2000): Primary 52C35, 14N20; Secondary 13N15, 15A36
Published electronically: March 20, 2007
MathSciNet review: 2309190
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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.


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Additional Information

Max Wakefield
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Address at time of publication: Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido 060-0810, Japan
Email: mwakefie@math.uoregon.edu

Sergey Yuzvinsky
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Email: yuz@math.uoregon.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04222-5
Received by editor(s): July 25, 2005
Received by editor(s) in revised form: September 8, 2005
Published electronically: March 20, 2007
Additional Notes: Research at MSRI was supported in part by NSF grant DMS-9810361
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.