*Paula Tretkoff*

- Published in print:
- 2016
- Published Online:
- October 2017
- ISBN:
- 9780691144771
- eISBN:
- 9781400881253
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691144771.003.0004
- Subject:
- Mathematics, Geometry / Topology

This chapter deals with complex surfaces and their finite coverings branched along divisors, that is, subvarieties of codimension 1. In particular, it considers coverings branched over transversally ...
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This chapter deals with complex surfaces and their finite coverings branched along divisors, that is, subvarieties of codimension 1. In particular, it considers coverings branched over transversally intersecting divisors. Applying this to linear arrangements in the complex projective plane, the chapter first blows up the projective plane at non-transverse intersection points, that is, at those points of the arrangement where more than two lines intersect. These points are called singular points of the arrangement. This gives rise to a complex surface and transversely intersecting divisors that contain the proper transforms of the original lines. The chapter also introduces the divisor class group, their intersection numbers, and the canonical divisor class. Finally, it describes the Chern numbers of a complex surface in order to define the proportionality deviation of a complex surface and to study its behavior with respect to finite covers.Less

This chapter deals with complex surfaces and their finite coverings branched along divisors, that is, subvarieties of codimension 1. In particular, it considers coverings branched over transversally intersecting divisors. Applying this to linear arrangements in the complex projective plane, the chapter first blows up the projective plane at non-transverse intersection points, that is, at those points of the arrangement where more than two lines intersect. These points are called singular points of the arrangement. This gives rise to a complex surface and transversely intersecting divisors that contain the proper transforms of the original lines. The chapter also introduces the divisor class group, their intersection numbers, and the canonical divisor class. Finally, it describes the Chern numbers of a complex surface in order to define the proportionality deviation of a complex surface and to study its behavior with respect to finite covers.

*Paula Tretkoff*

- Published in print:
- 2016
- Published Online:
- October 2017
- ISBN:
- 9780691144771
- eISBN:
- 9781400881253
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691144771.003.0006
- Subject:
- Mathematics, Geometry / Topology

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 ...
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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.Less

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.

*Richard Evan Schwartz*

- Published in print:
- 2019
- Published Online:
- September 2019
- ISBN:
- 9780691181387
- eISBN:
- 9780691188997
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691181387.003.0003
- Subject:
- Mathematics, Educational Mathematics

This chapter derives some basic properties of the plaid model. It is organized as follows. Section 2.2 deals with the symmetries of the plaid model. First, it deals with the unoriented model and then ...
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This chapter derives some basic properties of the plaid model. It is organized as follows. Section 2.2 deals with the symmetries of the plaid model. First, it deals with the unoriented model and then considers the oriented model. Section 2.3 proves the technical lemma that each unit integer segment contains exactly two intersection points. The work in this section reveals the nice geometric way that the slanting lines intersect each unit integer square. Section 2.4 establishes the following result: Within each block, there are exactly two lines of capacity k for each even k ɛ [0, ω]. Moreover, within the block, each line of capacity k has exactly k light points on it (when double points are appropriately counted). Section 2.5 establishes a subtle additional symmetry of the plaid model.Less

This chapter derives some basic properties of the plaid model. It is organized as follows. Section 2.2 deals with the symmetries of the plaid model. First, it deals with the unoriented model and then considers the oriented model. Section 2.3 proves the technical lemma that each unit integer segment contains exactly two intersection points. The work in this section reveals the nice geometric way that the slanting lines intersect each unit integer square. Section 2.4 establishes the following result: Within each block, there are exactly two lines of capacity *k* for each even *k* ɛ [0, ω]. Moreover, within the block, each line of capacity *k* has exactly *k* light points on it (when double points are appropriately counted). Section 2.5 establishes a subtle additional symmetry of the plaid model.