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Complex Ball Quotients and Line Arrangements in the Projective Plane (MN-51)$
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Paula Tretkoff

Print publication date: 2016

Print ISBN-13: 9780691144771

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691144771.001.0001

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Line Arrangements in P2(C) and Their Finite Covers

Line Arrangements in P2(C) and Their Finite Covers

(p.85) Chapter Five Line Arrangements in P2(C) and Their Finite Covers
Complex Ball Quotients and Line Arrangements in the Projective Plane (MN-51)

Paula Tretkoff

Princeton University Press

This chapter discusses the free 2-ball quotients arising as finite covers of the projective plane branched along line arrangements. It first considers a surface X obtained by blowing up the singular intersection points of a linear arrangement in the complex projective plane, as well as a smooth compact complex surface Y that is a finite covering of X. If Y is of general type with vanishing proportionality deviation, then it is a free 2-ball quotient. The chapter then looks at line arrangements that have equal ramification indices along each of the proper transforms of the original lines, along with cases of blowing down rational curves and removing elliptic curves. It also enumerates all possibilities for the assigned weights of the arrangements, under the assumption that divisors of negative or infinite weight on the blown-up line arrangements do not intersect.

Keywords:   projective plane, line arrangement, intersection point, complex surface, finite covering, proportionality deviation, free 2-ball quotient, ramification indices, rational curve, elliptic curve

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